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<div class="section" id="fwk-redden-ch05_s06" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">5.6</span> Solving Radical Equations</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch05_s06_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch05_s06_o01" numeration="arabic">
<li>Solve equations involving square roots.</li>
<li>Solve equations involving cube roots.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch05_s06_s01" version="5.0" lang="en">
<h2 class="title editable block">Radical Equations</h2>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p01">A <span class="margin_term"><a class="glossterm">radical equation</a><span class="glossdef">Any equation that contains one or more radicals with a variable in the radicand.</span></span> is any equation that contains one or more radicals with a variable in the radicand. Following are some examples of radical equations, all of which will be solved in this section:</p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p02">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="right"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1621" display="inline"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mn>3</mn></mrow></math></span></p></td>
<td align="right"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1622" display="inline"><mrow><mroot><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p></td>
<td align="right"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1623" display="inline"><mrow><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt><mo>−</mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mn>1</mn></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch05_s06_s01_p03">We begin with the <span class="margin_term"><a class="glossterm">squaring property of equality</a><span class="glossdef">Given real numbers <em class="emphasis">a</em> and <em class="emphasis">b</em>, where <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1624" display="inline"><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow></math></span>, then <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1625" display="inline"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></span></span>; given real numbers <em class="emphasis">a</em> and <em class="emphasis">b</em>, we have the following:</p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1626" display="block"><mrow><mtext>If</mtext><mtext> </mtext><mi>a</mi><mo>=</mo><mi>b</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mtext>then</mtext><mtext> </mtext><mtext> </mtext><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mo>.</mo></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p05">In other words, equality is retained if we square both sides of an equation.</p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1627" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mtext>−</mtext><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>3</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>⇒</mo></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mn>9</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>9</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p07">The converse, on the other hand, is not necessarily true,</p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1628" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mn>9</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>⇒</mo></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>≠</mo></mtd><mtd columnalign="left"><mrow><mn>3</mn><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p09">This is important because we will use this property to solve radical equations. Consider a very simple radical equation that can be solved by inspection,</p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p10"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1629" display="block"><mrow><msqrt><mi>x</mi></msqrt><mo>=</mo><mn>5</mn></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p11">Here we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1630" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>25</mn></mrow></math></span> is a solution. To solve this equation algebraically, make use of the squaring property of equality and the fact that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1631" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mi>a</mi></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msqrt><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></msqrt><mo>=</mo><mi>a</mi></mrow></math></span> when <em class="emphasis">a</em> is nonnegative. Eliminate the square root by squaring both sides of the equation as follows:</p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p12"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1632" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo color="#007fbf">(</mo><mrow><msqrt><mi>x</mi></msqrt></mrow><mo color="#007fbf">)</mo></mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo color="#007fbf">(</mo><mn>5</mn><mo color="#007fbf">)</mo></mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></msup></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>25</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p13">As a check, we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1633" display="inline"><mrow><msqrt><mrow><mn>25</mn></mrow></msqrt><mo>=</mo><mn>5</mn></mrow></math></span> as expected. Because the converse of the squaring property of equality is not necessarily true, solutions to the squared equation may not be solutions to the original. Hence squaring both sides of an equation introduces the possibility of <span class="margin_term"><a class="glossterm">extraneous solutions</a><span class="glossdef">A properly found solution that does not solve the original equation.</span></span>, which are solutions that do not solve the original equation. For example,</p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p14"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1634" display="block"><mrow><msqrt><mi>x</mi></msqrt><mo>=</mo><mtext>−</mtext><mn>5</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p15">This equation clearly does not have a real number solution. However, squaring both sides gives us a solution:</p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p16"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1635" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo color="#007fbf">(</mo><mrow><msqrt><mi>x</mi></msqrt></mrow><mo color="#007fbf">)</mo></mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo color="#007fbf">(</mo><mrow><mtext>−</mtext><mn>5</mn></mrow><mo color="#007fbf">)</mo></mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></msup></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>25</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p17">As a check, we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1636" display="inline"><mrow><msqrt><mrow><mn>25</mn></mrow></msqrt><mo>≠</mo><mo>−</mo><mn>5</mn></mrow><mo>.</mo></math></span> For this reason, we must check the answers that result from squaring both sides of an equation.</p>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p18">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1637" display="inline"><mrow><msqrt><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p19">We can eliminate the square root by applying the squaring property of equality.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p20"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1638" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>r</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>16</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>15</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p21">Next, we must check.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p22"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1639" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>5</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>15</mn><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>16</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>4</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p23">Answer: The solution is 5.</p>
</div>
<p class="para block" id="fwk-redden-ch05_s06_s01_p24">There is a geometric interpretation to the previous example. Graph the function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1640" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></math></span> and determine where it intersects the graph defined by <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1641" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large block">
<img src="section_08/3f037eb56381aa157ac81ee8bd6a85a8.png">
</div>
<p class="para block" id="fwk-redden-ch05_s06_s01_p26">As illustrated, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1642" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1643" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>5</mn></mrow><mo>.</mo></math></span></p>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p27">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1644" display="inline"><mrow><msqrt><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt><mo>=</mo><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p28">Begin by squaring both sides of the equation.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p29"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1645" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>r</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p30">The resulting quadratic equation can be solved by factoring.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p31"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1646" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>11</mn><mi>x</mi><mo>+</mo><mn>28</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>or</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p32">Checking the solutions after squaring both sides of an equation is not optional. Use the original equation when performing the check.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p33">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1647" display="inline"><mrow><mstyle color="#007fbf"><mrow><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mrow></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>4</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1648" display="inline"><mrow><mstyle color="#007fbf"><mrow><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mrow></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>7</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1649" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mstyle color="#007fbf"><mn>4</mn></mstyle><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mn>4</mn></mstyle><mo>−</mo><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mn>1</mn></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>1</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1650" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mstyle color="#007fbf"><mn>7</mn></mstyle><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mn>7</mn></mstyle><mo>−</mo><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mn>4</mn></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>2</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch05_s06_s01_p34">After checking, you can see that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1651" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>4</mn></mrow></math></span> is an extraneous solution; it does not solve the original radical equation. Disregard that answer. This leaves <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1652" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>7</mn></mrow></math></span> as the only solution.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p35">Answer: The solution is 7.</p>
</div>
<p class="para block" id="fwk-redden-ch05_s06_s01_p36">Geometrically we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1653" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msqrt></mrow></math></span> is equal to <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1654" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mo>−</mo><mn>5</mn></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1655" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>7</mn></mrow><mo>.</mo></math></span></p>
<div class="informalfigure large block">
<img src="section_08/2cf52a39d571cb46cefee616d835a8b4.png">
</div>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p38">In the previous two examples, notice that the radical is isolated on one side of the equation. Typically, this is not the case. The steps for solving radical equations involving square roots are outlined in the following example.</p>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p39">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1656" display="inline"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>+</mo><mn>2</mn><mo>=</mo><mi>x</mi></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p40"><strong class="emphasis bold">Step 1:</strong> Isolate the square root. Begin by subtracting 2 from both sides of the equation.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p41"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1657" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>+</mo><mn>2</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi></mtd></mtr><mtr><mtd columnalign="right"><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi><mo>−</mo><mn>2</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p42"><strong class="emphasis bold">Step 2:</strong> Square both sides. Squaring both sides eliminates the square root.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p43"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1658" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p44"><strong class="emphasis bold">Step 3:</strong> Solve the resulting equation. Here we are left with a quadratic equation that can be solved by factoring.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p45"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1659" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>or</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p46"><strong class="emphasis bold">Step 4:</strong> Check the solutions in the original equation. Squaring both sides introduces the possibility of extraneous solutions; hence the check is required.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p47">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1660" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1661" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>5</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1662" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>1</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></msqrt><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>1</mn></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mn>1</mn></msqrt><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>1</mn><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>3</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1663" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>5</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></msqrt><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>5</mn></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mn>9</mn></msqrt><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>3</mn><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>5</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch05_s06_s01_p48">After checking, we can see that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1664" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></math></span> is an extraneous solution; it does not solve the original radical equation. This leaves <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1665" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>5</mn></mrow></math></span> as the only solution.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p49">Answer: The solution is 5.</p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p50">Sometimes there is more than one solution to a radical equation.</p>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p51">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1666" display="inline"><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt><mo>−</mo><mi>x</mi><mo>=</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p52">Begin by isolating the term with the radical.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p53"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1667" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt><mo>−</mo><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>d</mi><mi>d</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>t</mi><mi>o</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p54">Despite the fact that the term on the left side has a coefficient, we still consider it to be isolated. Recall that terms are separated by addition or subtraction operators.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p55"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1668" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>r</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p56">Solve the resulting quadratic equation.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p57"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1669" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>8</mn><mi>x</mi><mo>+</mo><mn>20</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>or</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>2</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p58">Since we squared both sides, we must check our solutions.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p59">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1670" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mo>−</mo><mn>2</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1671" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>2</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1672" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt><mo>−</mo><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>5</mn></mrow></msqrt><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>2</mn></mstyle></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mrow><mo>−</mo><mn>4</mn><mo>+</mo><mn>5</mn></mrow></msqrt><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mn>1</mn></msqrt><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>4</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1673" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt><mo>−</mo><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>5</mn></mrow></msqrt><mo>−</mo><mrow><mo>(</mo><mstyle color="#007fbf"><mn>2</mn></mstyle><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mrow><mn>4</mn><mo>+</mo><mn>5</mn></mrow></msqrt><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msqrt><mn>9</mn></msqrt><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>6</mn><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>4</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch05_s06_s01_p60">After checking, we can see that both are solutions to the original equation.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p61">Answer: The solutions are ±2.</p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p62">Sometimes both of the possible solutions are extraneous.</p>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n05">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p63">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1674" display="inline"><mrow><msqrt><mrow><mn>4</mn><mo>−</mo><mn>11</mn><mi>x</mi></mrow></msqrt><mo>−</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p64">Begin by isolating the radical.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p65"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1675" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>4</mn><mo>−</mo><mn>11</mn><mi>x</mi></mrow></msqrt><mo>−</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>I</mi><mi>s</mi><mi>o</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>4</mn><mo>−</mo><mn>11</mn><mi>x</mi></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mrow><mn>4</mn><mo>−</mo><mn>11</mn><mi>x</mi></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>r</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><mo>−</mo><mn>11</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p66"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1676" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>7</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p67">Since we squared both sides, we must check our solutions.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p68">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1677" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1678" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mo>−</mo><mn>7</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1679" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>4</mn><mo>−</mo><mn>11</mn><mi>x</mi></mrow></msqrt><mo>−</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>4</mn><mo>−</mo><mn>11</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>)</mo></mrow></mrow></msqrt><mo>−</mo><mstyle color="#007fbf"><mn>0</mn></mstyle><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mn>4</mn></msqrt><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>4</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1680" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>4</mn><mo>−</mo><mn>11</mn><mi>x</mi></mrow></msqrt><mo>−</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>4</mn><mo>−</mo><mn>11</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>7</mn></mstyle></mrow><mo>)</mo></mrow></mrow></msqrt><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>7</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>4</mn><mo>+</mo><mn>77</mn></mrow></msqrt><mo>+</mo><mn>7</mn><mo>+</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>81</mn></mrow></msqrt><mo>+</mo><mn>9</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>9</mn><mo>+</mo><mn>9</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>18</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch05_s06_s01_p69">Since both possible solutions are extraneous, the equation has no solution.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p70">Answer: No solution, Ø</p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p71">The squaring property of equality extends to any positive integer power <em class="emphasis">n</em>. Given real numbers <em class="emphasis">a</em> and <em class="emphasis">b</em>, we have the following:</p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p72"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1681" display="block"><mrow><mtext>If</mtext><mtext> </mtext><mi>a</mi><mo>=</mo><mi>b</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mtext>then</mtext><mtext> </mtext><mtext> </mtext><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><msup><mi>b</mi><mi>n</mi></msup><mo>.</mo></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p73">This is often referred to as the <span class="margin_term"><a class="glossterm">power property of equality</a><span class="glossdef">Given any positive integer <em class="emphasis">n</em> and real numbers <em class="emphasis">a</em> and <em class="emphasis">b</em> where <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1682" display="inline"><mrow><mi>a</mi><mo>=</mo><mi>b</mi></mrow></math></span>, then <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1683" display="inline"><mrow><msup><mi>a</mi><mi>n</mi></msup><mo>=</mo><msup><mi>b</mi><mi>n</mi></msup></mrow><mo>.</mo></math></span></span></span>. Use this property, along with the fact that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1684" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mroot><mi>a</mi><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow><mo>)</mo></mrow></mrow><mi>n</mi></msup><mo>=</mo><mroot><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow><mpadded width="0.4em"><mi>n</mi></mpadded></mroot><mo>=</mo><mi>a</mi></mrow></math></span>, when <em class="emphasis">a</em> is nonnegative, to solve radical equations with indices greater than 2.</p>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n06">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p74">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1685" display="inline"><mrow><mroot><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p75">Isolate the radical, and then cube both sides of the equation.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p76"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1686" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mroot><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>I</mi><mi>s</mi><mi>o</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>a</mi><mi>d</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi><mo>.</mo></mrow></mstyle></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mroot><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mroot><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>C</mi><mi>u</mi><mi>b</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>S</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>1</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p77">Check.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p78">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1687" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p></th>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1688" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1689" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow><mn>3</mn></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow><mn>3</mn></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>4</mn><mo>⋅</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mn>7</mn></mrow><mn>3</mn></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>1</mn><mo>+</mo><mn>7</mn></mrow><mn>3</mn></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mn>8</mn><mn>3</mn></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1690" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow><mn>3</mn></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>7</mn></mrow><mn>3</mn></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>4</mn><mo>⋅</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mn>7</mn></mrow><mn>3</mn></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>1</mn><mo>+</mo><mn>7</mn></mrow><mn>3</mn></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mn>8</mn><mn>3</mn></mroot><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>0</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch05_s06_s01_p79">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1691" display="inline"><mrow><mo>±</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span>.</p>
</div>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n06a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p80"><strong class="emphasis bold">Try this!</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1692" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>3</mn><msqrt><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mn>3</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p81">Answer: The solution is 33.</p>
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</div>
</div>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p83">It may be the case that the equation has more than one term that consists of radical expressions.</p>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n07">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p84">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1693" display="inline"><mrow><msqrt><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p85">Both radicals are considered isolated on separate sides of the equation.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p86"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1694" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>r</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p87">Check <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1695" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>2</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p88"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1696" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mn>2</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>10</mn><mo>−</mo><mn>3</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>8</mn><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mn>7</mn></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mn>7</mn></msqrt><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p89">Answer: The solution is 2.</p>
</div>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n08">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p90">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1697" display="inline"><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>14</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><mi>x</mi><mo>+</mo><mn>50</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p91">Eliminate the radicals by cubing both sides.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p92"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1698" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>14</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mi>x</mi><mo>+</mo><mn>50</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>14</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mroot><mrow><mi>x</mi><mo>+</mo><mn>50</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>C</mi><mi>u</mi><mi>b</mi><mi>e</mi><mtext> </mtext><mi>b</mi><mi>o</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>s</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>s</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>14</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>50</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>S</mi><mi>o</mi><mi>l</mi><mi>v</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>or</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>8</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p93">Check.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p94">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1699" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mo>−</mo><mn>8</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1700" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>8</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1701" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>14</mn></mrow><mn>3</mn></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mi>x</mi><mo>+</mo><mn>50</mn></mrow><mn>3</mn></mroot></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>8</mn></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>8</mn></mstyle></mrow><mo>)</mo></mrow><mo>−</mo><mn>14</mn></mrow><mn>3</mn></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>8</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>50</mn></mrow><mn>3</mn></mroot></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>64</mn><mo>−</mo><mn>8</mn><mo>−</mo><mn>14</mn></mrow><mn>3</mn></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mn>42</mn></mrow><mn>3</mn></mroot></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>42</mn></mrow><mn>3</mn></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mn>42</mn></mrow><mn>3</mn></mroot><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1702" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>14</mn></mrow><mn>3</mn></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mi>x</mi><mo>+</mo><mn>50</mn></mrow><mn>3</mn></mroot></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><msup><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mn>8</mn></mstyle><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mrow><mo>(</mo><mstyle color="#007fbf"><mn>8</mn></mstyle><mo>)</mo></mrow><mo>−</mo><mn>14</mn></mrow><mn>3</mn></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mn>8</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>50</mn></mrow><mn>3</mn></mroot></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>64</mn><mo>+</mo><mn>8</mn><mo>−</mo><mn>14</mn></mrow><mn>3</mn></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mn>58</mn></mrow><mn>3</mn></mroot></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mroot><mrow><mn>58</mn></mrow><mn>3</mn></mroot></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mroot><mrow><mn>58</mn></mrow><mn>3</mn></mroot><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch05_s06_s01_p95">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1703" display="inline"><mrow><mo>±</mo><mn>8</mn></mrow><mo>.</mo></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p96">It may not be possible to isolate a radical on both sides of the equation. When this is the case, isolate the radicals, one at a time, and apply the squaring property of equality multiple times until only a polynomial remains.</p>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n09">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p97">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1704" display="inline"><mrow><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt><mo>−</mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mn>1</mn></mrow></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p98">Begin by isolating one of the radicals. In this case, add <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1705" display="inline"><mrow><msqrt><mi>x</mi></msqrt></mrow></math></span> to both sides of the equation.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p99"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1706" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt><mo>−</mo><msqrt><mi>x</mi></msqrt></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>1</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p100">Next, square both sides. Take care to apply the distributive property to the right side.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p101"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1707" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>2</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>2</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt><mo>+</mo><msqrt><mi>x</mi></msqrt><mo>+</mo><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>2</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi><mo>+</mo><mn>2</mn><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>1</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p102">At this point we have one term that contains a radical. Isolate it and square both sides again.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p103"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1708" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mn>2</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi><mo>+</mo><mn>2</mn><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>1</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><msqrt><mi>x</mi></msqrt></mtd></mtr><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mrow><mn>2</mn><msqrt><mi>x</mi></msqrt></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"><mn>1</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mi>x</mi></mtd></mtr><mtr><mtd columnalign="right"><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p104">Check to see if <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1709" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span> satisfies the original equation <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1710" display="inline"><mrow><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt><mo>−</mo><msqrt><mi>x</mi></msqrt><mo>=</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p105"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1711" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mstyle color="#007f3f"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mstyle><mo>+</mo><mn>2</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mstyle color="#007f3f"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mstyle></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></msqrt><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>2</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>1</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p106">Answer: The solution is <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1712" display="inline"><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>.</mo></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch05_s06_s01_p107"><em class="emphasis bolditalic">Note:</em> Because <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1713" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>≠</mo><msup><mi>A</mi><mn>2</mn></msup><mo>+</mo><msup><mi>B</mi><mn>2</mn></msup></mrow></math></span>, we cannot simply square each term. For example, it is incorrect to square each term as follows.</p>
<p class="para block" id="fwk-redden-ch05_s06_s01_p108"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1714" display="block"><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><msup><mrow><mo color="#007fbf">(</mo><mrow><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt></mrow><mo color="#007fbf">)</mo></mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></msup><mo>−</mo><msup><mrow><mo color="#007fbf">(</mo><mrow><msqrt><mi>x</mi></msqrt></mrow><mo color="#007fbf">)</mo></mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></msup><mo>=</mo><msup><mrow><mo color="#007fbf">(</mo><mn>1</mn><mo color="#007fbf">)</mo></mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></msup></mtd></mtr><mtr columnalign="left"><mtd columnalign="center"><mstyle color="#ff0000"><mi>I</mi><mi>n</mi><mi>c</mi><mi>o</mi><mi>r</mi><mi>r</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>!</mi></mstyle></mtd></mtr></mtable></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s06_s01_p109">This is a common mistake and leads to an incorrect result. When squaring both sides of an equation with multiple terms, we must take care to apply the distributive property.</p>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n10">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p110">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1715" display="inline"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt><mo>=</mo><mn>1</mn></mrow></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p111">Begin by isolating one of the radicals. In this case, add <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1716" display="inline"><mrow><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt></mrow></math></span> to both sides of the equation.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p112"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1717" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></msqrt></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt><mo>+</mo><mn>1</mn></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p113">Next, square both sides. Take care to apply the distributive property to the right side.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p114"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1718" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mrow><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi><mo>+</mo><mn>6</mn><mo>+</mo><mn>2</mn><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi><mo>+</mo><mn>7</mn><mo>+</mo><mn>2</mn><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p115">At this point we have one term that contains a radical. Isolate it and square both sides again.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p116"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1719" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>x</mi><mo>+</mo><mn>7</mn><mo>+</mo><mn>2</mn><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>24</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>15</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mtext>or</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>5</mn></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p117">Check.</p>
<p class="para" id="fwk-redden-ch05_s06_s01_p118">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1720" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mn>3</mn></mrow></math></span></p></th>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1721" display="inline"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mstyle><mtext> </mtext><mi>x</mi><mo>=</mo><mo>−</mo><mn>5</mn></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1722" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007fbf"><mn>3</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>10</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mstyle color="#007fbf"><mn>3</mn></mstyle><mo>+</mo><mn>6</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>16</mn></mrow></msqrt><mo>−</mo><msqrt><mn>9</mn></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><mo>−</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>1</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1723" display="inline"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>5</mn></mstyle></mrow><mo>)</mo></mrow><mo>+</mo><mn>10</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mstyle color="#007fbf"><mo>−</mo><mn>5</mn></mstyle><mo>+</mo><mn>6</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mn>0</mn></msqrt><mo>−</mo><msqrt><mn>1</mn></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>0</mn><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>−</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch05_s06_s01_p119">Answer: The solution is 3.</p>
</div>
<div class="callout block" id="fwk-redden-ch05_s06_s01_n10a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch05_s06_s01_p120"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1724" display="inline"><mrow><msqrt><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>21</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>22</mn></mrow></msqrt><mo>=</mo><mn>1</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch05_s06_s01_p121">Answer: The solution is 7.</p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/BtMeQWPsMaY" condition="http://img.youtube.com/vi/BtMeQWPsMaY/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/BtMeQWPsMaY" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<div class="key_takeaways editable block" id="fwk-redden-ch05_s06_s01_n11">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch05_s06_s01_l01" mark="bullet">
<li>Solve equations involving square roots by first isolating the radical and then squaring both sides. Squaring a square root eliminates the radical, leaving us with an equation that can be solved using the techniques learned earlier in our study of algebra.</li>
<li>Squaring both sides of an equation introduces the possibility of extraneous solutions. For this reason, you must check your solutions in the original equation.</li>
<li>Solve equations involving <em class="emphasis">n</em>th roots by first isolating the radical and then raise both sides to the <em class="emphasis">n</em>th power. This eliminates the radical and results in an equation that may be solved with techniques you have already mastered.</li>
<li>When more than one radical term is present in an equation, isolate them one at a time, and apply the power property of equality multiple times until only a polynomial remains.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch05_s06_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s06_qs01_qd01">
<h3 class="title">Part A: Solving Radical Equations</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s06_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch05_s06_qs01_p01"><strong class="emphasis bold">Solve</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1725" display="inline"><mrow><msqrt><mi>x</mi></msqrt><mo>=</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1726" display="inline"><mrow><msqrt><mi>x</mi></msqrt><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1727" display="inline"><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>8</mn><mo>=</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1728" display="inline"><mrow><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>4</mn><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1729" display="inline"><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>7</mn><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1730" display="inline"><mrow><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>3</mn><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1731" display="inline"><mrow><mn>5</mn><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1733" display="inline"><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1734" display="inline"><mrow><msqrt><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1735" display="inline"><mrow><msqrt><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></msqrt><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p22"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1736" display="inline"><mrow><msqrt><mrow><mn>7</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msqrt><mo>+</mo><mn>6</mn><mo>=</mo><mn>11</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1737" display="inline"><mrow><msqrt><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></msqrt><mo>+</mo><mn>9</mn><mo>=</mo><mn>14</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p26"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1738" display="inline"><mrow><mn>2</mn><msqrt><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1740" display="inline"><mrow><mn>3</mn><msqrt><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1741" display="inline"><mrow><msqrt><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>=</mo><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1742" display="inline"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mn>2</mn><mi>x</mi></mrow></msqrt><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1744" display="inline"><mrow><msqrt><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mn>2</mn><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1746" display="inline"><mrow><msqrt><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>11</mn></mrow></msqrt><mo>=</mo><mn>2</mn><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1747" display="inline"><mrow><msqrt><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></msqrt><mo>=</mo><msqrt><mi>x</mi></msqrt><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1748" display="inline"><mrow><msqrt><mrow><mn>25</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mn>5</mn><msqrt><mi>x</mi></msqrt><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1749" display="inline"><mrow><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1750" display="inline"><mrow><mroot><mi>x</mi><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mtext>−</mtext><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p46"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1751" display="inline"><mrow><mroot><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p48"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1752" display="inline"><mrow><mroot><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>11</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p50"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1753" display="inline"><mrow><mroot><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>3</mn><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p52"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1754" display="inline"><mrow><mroot><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>5</mn><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p54"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1755" display="inline"><mrow><mn>4</mn><mo>−</mo><mn>2</mn><mroot><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1756" display="inline"><mrow><mn>6</mn><mo>−</mo><mn>3</mn><mroot><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1758" display="inline"><mrow><mroot><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p60"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1760" display="inline"><mrow><mroot><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>+</mo><mn>5</mn><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p62"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1761" display="inline"><mrow><msqrt><mrow><mn>8</mn><mi>x</mi><mo>+</mo><mn>11</mn></mrow></msqrt><mo>=</mo><mn>3</mn><msqrt><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p64"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1762" display="inline"><mrow><mn>2</mn><msqrt><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p66"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1763" display="inline"><mrow><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow></msqrt><mo>=</mo><msqrt><mrow><mn>7</mn><mi>x</mi><mo>−</mo><mn>15</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p68"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1764" display="inline"><mrow><msqrt><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></msqrt><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p70"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1765" display="inline"><mrow><mroot><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><mn>4</mn><mi>x</mi></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p72"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1766" display="inline"><mrow><mroot><mrow><mn>9</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p74"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1767" display="inline"><mrow><mroot><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p76"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1768" display="inline"><mrow><mroot><mrow><mn>9</mn><mi>x</mi></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><mn>3</mn><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>6</mn><mo stretchy="false">)</mo></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p78"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1769" display="inline"><mrow><mroot><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>=</mo><mroot><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p80"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1770" display="inline"><mrow><mroot><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>=</mo><mroot><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p82"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1771" display="inline"><mrow><msqrt><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>21</mn></mrow></msqrt><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p84"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1772" display="inline"><mrow><msqrt><mrow><mn>8</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow></msqrt><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p86"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1773" display="inline"><mrow><msqrt><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></msqrt><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p88"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1774" display="inline"><mrow><msqrt><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow></msqrt><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p90"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1775" display="inline"><mrow><mn>2</mn><msqrt><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p92"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1776" display="inline"><mrow><mn>3</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>9</mn></mrow></msqrt><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p94"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1777" display="inline"><mrow><msqrt><mrow><mn>9</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow></msqrt><mo>=</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p96"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1778" display="inline"><mrow><msqrt><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></msqrt><mo>=</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p98"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1779" display="inline"><mrow><msqrt><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p100"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1780" display="inline"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></msqrt><mo>=</mo><mi>x</mi><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p102"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1781" display="inline"><mrow><msqrt><mrow><mn>16</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow></msqrt><mo>=</mo><mi>x</mi><mo>−</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p104"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1782" display="inline"><mrow><msqrt><mrow><mn>7</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow></msqrt><mo>=</mo><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p106"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1783" display="inline"><mrow><mn>3</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow></msqrt><mo>=</mo><mi>x</mi><mo>+</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p108"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1784" display="inline"><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt><mo>=</mo><mi>x</mi><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1785" display="inline"><mrow><mn>3</mn><msqrt><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>−</mo><mn>1</mn><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p112"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1786" display="inline"><mrow><mn>2</mn><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt><mo>−</mo><mn>1</mn><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p114"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1787" display="inline"><mrow><msqrt><mrow><mn>10</mn><mi>x</mi><mo>+</mo><mn>41</mn></mrow></msqrt><mo>−</mo><mn>5</mn><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1788" display="inline"><mrow><msqrt><mrow><mn>6</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></msqrt><mo>−</mo><mn>3</mn><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p118"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1789" display="inline"><mrow><msqrt><mrow><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mn>2</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p120"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1791" display="inline"><mrow><msqrt><mrow><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mn>3</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1793" display="inline"><mrow><mn>5</mn><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt><mo>=</mo><mi>x</mi><mo>+</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1794" display="inline"><mrow><mn>4</mn><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></msqrt><mo>=</mo><mi>x</mi><mo>+</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p126"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1795" display="inline"><mrow><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn></mrow></msqrt><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1796" display="inline"><mrow><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn></mrow></msqrt><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p130"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1797" display="inline"><mrow><mn>3</mn><mo>+</mo><msqrt><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>11</mn></mrow></msqrt><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p132"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1798" display="inline"><mrow><mn>2</mn><mo>+</mo><msqrt><mrow><mn>9</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow></msqrt><mo>=</mo><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p134"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1799" display="inline"><mrow><msqrt><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></msqrt><mo>−</mo><mi>x</mi><mo>=</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p136"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1800" display="inline"><mrow><msqrt><mrow><mn>8</mn><mi>x</mi><mo>+</mo><mn>73</mn></mrow></msqrt><mo>−</mo><mi>x</mi><mo>=</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p138"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1801" display="inline"><mrow><mn>2</mn><msqrt><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msqrt><mo>−</mo><mn>3</mn><mo>=</mo><mn>2</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p140"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1803" display="inline"><mrow><mn>2</mn><msqrt><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msqrt><mo>−</mo><mn>3</mn><mo>=</mo><mn>3</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p142"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1805" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>4</mn><mo>=</mo><msqrt><mrow><mn>14</mn><mo>−</mo><mn>10</mn><mi>x</mi></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p144"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1806" display="inline"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>6</mn><mo>=</mo><msqrt><mrow><mn>33</mn><mo>−</mo><mn>24</mn><mi>x</mi></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p146"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1807" display="inline"><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>24</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p148"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1808" display="inline"><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>54</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p150"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1809" display="inline"><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>1</mn><mo>=</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p152"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1810" display="inline"><mrow><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>5</mn><mo>=</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p154"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1811" display="inline"><mrow><mroot><mrow><mn>25</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mtext>−</mtext><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p156"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1813" display="inline"><mrow><mroot><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>23</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mtext>−</mtext><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p158"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1814" display="inline"><mrow><mroot><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p160"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1816" display="inline"><mrow><mn>4</mn><mroot><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p162"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1818" display="inline"><mrow><mroot><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p164"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1819" display="inline"><mrow><mroot><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>20</mn><mi>x</mi></mrow><mpadded width="0.4em"><mn>5</mn></mpadded></mroot><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p166"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1821" display="inline"><mrow><msqrt><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>15</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p168"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1822" display="inline"><mrow><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mi>x</mi></mrow><mo>)</mo></mrow></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p170"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1824" display="inline"><mrow><mroot><mrow><mn>2</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>20</mn></mrow><mo>)</mo></mrow></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p172"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1825" display="inline"><mrow><mroot><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>40</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>=</mo><mroot><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p174"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1827" display="inline"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>2</mn><mi>x</mi></mrow></msqrt><mo>=</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p176"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1829" display="inline"><mrow><msqrt><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>13</mn></mrow></msqrt><mo>−</mo><mn>2</mn><msqrt><mi>x</mi></msqrt><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p178"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1831" display="inline"><mrow><msqrt><mrow><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn></mrow></msqrt><mo>−</mo><mn>2</mn><msqrt><mrow><mn>2</mn><mo>−</mo><mi>x</mi></mrow></msqrt><mo>=</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p180"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1832" display="inline"><mrow><msqrt><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p182"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1833" display="inline"><mrow><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></msqrt><mo>−</mo><msqrt><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p184"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1834" display="inline"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></msqrt><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p186"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1835" display="inline"><mrow><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></msqrt><mo>−</mo><msqrt><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></msqrt><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p188"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1836" display="inline"><mrow><msqrt><mrow><mn>6</mn><mo>−</mo><mn>5</mn><mi>x</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>3</mn><mo>−</mo><mn>3</mn><mi>x</mi></mrow></msqrt><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p190"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1837" display="inline"><mrow><msqrt><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></msqrt><mo>−</mo><mn>1</mn><mo>=</mo><msqrt><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p192"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1838" display="inline"><mrow><msqrt><mrow><mn>14</mn><mo>−</mo><mn>11</mn><mi>x</mi></mrow></msqrt><mo>+</mo><msqrt><mrow><mn>7</mn><mo>−</mo><mn>9</mn><mi>x</mi></mrow></msqrt><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1839" display="inline"><mrow><msqrt><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>=</mo><msqrt><mn>3</mn></msqrt><mo>−</mo><msqrt><mrow><mn>2</mn><mo>−</mo><mi>x</mi></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p196"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1840" display="inline"><mrow><msqrt><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p198"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1841" display="inline"><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>10</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p200"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1842" display="inline"><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa101">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p202"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1843" display="inline"><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa102">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p204"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1844" display="inline"><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa103">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p206"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1845" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa104">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p208"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1846" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p210"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1847" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p212"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1848" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p214"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1849" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>15</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p216"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1851" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p218"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1853" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mi>x</mi><mo>=</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa110">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p220"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1854" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>36</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mi>x</mi><mo>=</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa111">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p222"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1855" display="inline"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>26</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mi>x</mi><mo>+</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa112">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p224"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1856" display="inline"><mrow><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa113">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p226"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1857" display="inline"><mrow><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa114">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p228"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1858" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa115">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p230"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1860" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa116">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p232"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1861" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s06_qs01_qd01_qd02" start="117">
<p class="para" id="fwk-redden-ch05_s06_qs01_p234"><strong class="emphasis bold">Determine the roots of the given functions. Recall that a root is a value in the domain that results in zero. In other words, find</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1862" display="inline"><mi>x</mi></math></span> <strong class="emphasis bold">where</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1863" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa117">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p235"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1864" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></msqrt><mo>−</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa118">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p237"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1865" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></msqrt><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa119">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p239"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1866" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msqrt><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></msqrt><mo>−</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa120">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p241"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1867" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>3</mn><msqrt><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow></msqrt><mo>−</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa121">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p243"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1868" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mroot><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s06_qs01_qa122">
<div class="question">
<p class="para" id="fwk-redden-ch05_s06_qs01_p245"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1869" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mroot><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mpadded width="0.4em"><mn>3</mn></mpadded></mroot><mo>+</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s06_qs01_qd01_qd03" start="123">