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2659adfb-00ad-44ac-9082-cae39c674a93.html
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<h4>Activity</h4>
<br>
<div class="os-raise-familysupport">
<p>Access the <a href="https://k12.openstax.org/contents/raise/resources/ad776e2f4b03a08bf69ddddc59a1efd31551b1a0" target="_blank">Desmos guide PDF</a> for tips on solving problems with the Desmos graphing calculator.</p>
</div>
<br>
<p>Here is an example of an equation being solved by graphing and by <span class="os-raise-ib-tooltip" data-schema-version="1.0" data-store="glossary-tooltip">completing the square</span>.</p>
<p><img height="171" src="https://k12.openstax.org/contents/raise/resources/f2655f0650fd05c200ff9c432bafd8068e71a420" width="268"></p>
<p>\(\begin{aligned} x^{2}+6 x+7 &=0 \\ x^{2}+6 x+9 &=2 \\ (x+3)^{2} &=2 \\ x+3 &=\pm \sqrt{2} \\ x &=-3 \pm \sqrt{2} \end{aligned}\) <br>
</p>
<p>Verify: \(\sqrt{2}\) is approximately 1.414. So \(-3+ \sqrt{2} \approx -1.586\) and \(-3- \sqrt{2} \approx -4.414\).</p>
<p>For each equation, find the exact solutions by completing the square and the approximate solutions by graphing. Then, verify that the solutions found using the two methods are close. If you get stuck, refer back to the example.</p>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1a" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<li>\(x^2+4x+1=0\)</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>Find the exact solutions by completing the square.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1a">
<p>Compare your answer:</p>
<p>\(\begin{aligned} x^{2}+4 x+1 &=0 \\ x^{2}+4 x &=-1 \\ x^{2}+4 x+4 &=-1+4 \\ x^{2}+4 x+4 &=3 \\ (x+2)^{2} &=3 \\ \sqrt{(x+2)^{2}} &=\sqrt{3} \\ x+2 &=\pm \sqrt{3} \\x&=-2 \pm \sqrt{3} \end{aligned}\) <br>
</p>
</div>
<br>
<ol class="os-raise-noindent" start="2" type="a">
<li>Graph the equation to find approximate solutions. </li>
</ol>
<p>Use the Desmos graphing tool or technology outside the course. </p>
<br>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1b" data-schema-version="1.0">
<div class="os-raise-ib-cta-content"> </div>
<div class="os-raise-ib-cta-prompt">
<p>When you have finished graphing, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1b">
<p>Compare your answer:</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="244" role="presentation" src="https://k12.openstax.org/contents/raise/resources/fc7c976f9f84826aaf037e005a96b757fba599c6" width="244"></p>
</div>
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="8598f8e0-fb62-4e5e-bf45-9db30d216c6b" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q1c-->
<div class="os-raise-ib-pset-problem" data-content-id="e4b9dc8d-0ea3-4566-af06-451d36647eee" data-problem-type="dropdown" data-solution="Yes" data-solution-options='["Yes", "No"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="3" type="a">
<li>Are the solutions you found using the two methods close? </li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Yes. Solutions were the same: \(x \approx -0.268\) and \(x \approx -3.732\).</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Incorrect. The correct answer is Yes. Solutions were the same: \(x \approx -0.268\) and \(x \approx -3.732\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2a" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li>\(x^2−10x+18=0\)</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>Find the exact solutions by completing the square.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal2a">
<p>Compare your answer:</p>
<p>\(\begin{aligned} x^{2}-10 x+18 &=0 \\ x^{2}-10 x &=-18 \\ x^{2}-10 x+25 &=-18+25 \\ x^{2}-10 x+25 &=7 \\ (x-5)^{2} &=7 \\ \sqrt{(x-5)^{2}} &=\sqrt{7} \\ x-5 &=\pm \sqrt{7} \\ x &=5 \pm \sqrt{7} \end{aligned}\) <br>
</p>
</div>
<br>
<ol class="os-raise-noindent" start="2" type="a">
<li>Graph the equation to find approximate solutions. </li>
</ol>
<p>Use the Desmos graphing tool or technology outside the course. </p>
<br>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2b" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
</div>
<div class="os-raise-ib-cta-prompt">
<p>When you have finished graphing, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal2b">
<p>Compare your answer:</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="300" role="presentation" src="https://k12.openstax.org/contents/raise/resources/e68253272f86e5daa2ea0ff1148ead33a54fd788" width="300"></p>
</div>
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="3b59c9ac-6cf0-40c5-8cf2-95d34eb9fde0" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q2c-->
<div class="os-raise-ib-pset-problem" data-content-id="0f0d59d2-2fcd-4daf-ac6a-c1ab9c931506" data-problem-type="dropdown" data-solution="Yes" data-solution-options='["Yes", "No"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="3" type="a">
<li>Are the solutions you found using the two methods close? </li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Yes. Solutions were the same: \(x \approx 7.646\) and \(x \approx 2.354\).</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Incorrect. The correct answer is Yes. \(x \approx 7.646\) and \(x \approx 2.354\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal3a" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="3">
<li>\(x^2+5x+\frac{1}{4}=0\)</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>Find the exact solutions by completing the square.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal3a">
<p>Compare your answer:</p>
<p>\(\begin{array}{rcl} x^{2}+5 x+\frac{1}{4} &=& 0 \\ x^{2}+5 x &=& -\frac{1}{4} \\ x^{2}+5 x+\left(\frac{5}{2}\right)^{2} &=& -\frac{1}{4}+\left(\frac{5}{2}\right)^{2} \\ x^{2}+5 x+\frac{25}{4} &=& -\frac{1}{4}+\frac{25}{4} \\ x^{2}+5 x+\frac{25}{4} &=& \frac{24}{4} \\ x^{2}+5 x+\frac{25}{4} &=& 6 \\ \left(x+\frac{5}{2}\right)^{2} &=& 6 \\ \sqrt{\left(x+\frac{5}{2}\right)^{2}} &=& \sqrt{6} \\ x+\frac{5}{2} &=& \pm \sqrt{6} \\ x &=& -\frac{5}{2} \pm \sqrt{6} \end{array}\) </p>
</div>
<br>
<ol class="os-raise-noindent" start="2" type="a">
<li>Graph the equation to find approximate solutions. </li>
</ol>
<p>Use the Desmos graphing tool or technology outside the course. </p>
<br>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal3b" data-schema-version="1.0">
<div class="os-raise-ib-cta-content"> </div>
<div class="os-raise-ib-cta-prompt">
<p>When you have finished graphing, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal3b">
<p>Compare your answer:</p>
<img alt height="310" role="presentation" src="https://k12.openstax.org/contents/raise/resources/78c3ebc4c9c1e4fcbe7e9c7c06eea1266ec14713" width="300">
</div>
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="b67862a2-4253-47c5-9ca2-8dee5d31972e" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q3c-->
<div class="os-raise-ib-pset-problem" data-content-id="e91e1f89-ee24-4fd8-a748-9709bcd4de75" data-problem-type="dropdown" data-solution="Yes" data-solution-options='["Yes", "No"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="3" type="a">
<li>Are the solutions you found using the two methods close?</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Yes. Solutions were the same: \(x \approx -4.949\) and \(x \approx -0.051\).</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Incorrect. The correct answer is Yes. \(x \approx -4.949\) and \(x \approx -0.051\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal4a" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="4">
<li>\(x^2+\frac{8}{3}x+\frac{14}{9}=0\)</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>Find the exact solutions by completing the square.</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal4a">
<p>Compare your answer:</p>
<p>\(\begin{array}{rcl} x^{2}+\frac{8}{3} x+\frac{14}{9} &=&0 \\ x^{2}+\frac{8}{3} x &=&-\frac{14}{9} \\ x^{2}+\frac{8}{3} x+\left(\frac{1}{2} \cdot \frac{8}{3}\right)^{2} &=&-\frac{14}{9}+\left(\frac{1}{2} \cdot \frac{8}{3}\right)^{2} \\ x^{2}+\frac{8}{3} x+\left(\frac{4}{3}\right)^{2} &=&-\frac{14}{9}+\left(\frac{4}{3}\right)^{2} \\ x^{2}+\frac{8}{3} x+\frac{16}{9} &=&-\frac{14}{9}+\frac{16}{9} \\ x^{2}+\frac{8}{3} x+\frac{16}{9} &=&\frac{2}{9} \\ \left(x+\frac{4}{3}\right)^{2} &=&\frac{2}{9} \\ \sqrt{\left(x+\frac{4}{3}\right)^{2}} &=&\sqrt{\frac{2}{9}} \\ x+\frac{4}{3} &=&\pm \frac{\sqrt{2}}{3} \\ x &=&-\frac{4}{3} \pm \frac{\sqrt{2}}{3} \end{array}\)</p>
</div>
<br>
<ol class="os-raise-noindent" start="2" type="a">
<li>Graph the equation to find approximate solutions. </li>
</ol>
<p>Use the Desmos graphing tool or technology outside the course. </p>
<br>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content"> </div>
<div class="os-raise-ib-cta-prompt">
<p>When you have finished graphing, then select the <strong>solution</strong> button to compare your work.</p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<p><img alt height="304" role="presentation" src="https://k12.openstax.org/contents/raise/resources/d3b8cf6ebb398802e31d5e1363326e90518c97ed" width="300"></p>
</div>
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="764c4320-d781-4ff8-a1b8-2157515168a1" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q4c-->
<div class="os-raise-ib-pset-problem" data-content-id="b14e169f-7b9e-4e99-ac18-2c18cbeddb81" data-problem-type="dropdown" data-solution="Yes" data-solution-options='["Yes", "No"]'>
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent" start="3" type="a">
<li>Are the solutions you found using the two methods close?</li>
</ol>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! Yes. Solutions were the same: \(x \approx -0.862\) and \(x \approx -1.805\).</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Incorrect. The correct answer is Yes. \(x \approx -0.862\) and \(x \approx -1.805\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<br>
<br>
<!-- "Start "Are you Ready for More? click to reveal. If you have more than one on a page, you will need to change the Data-fire/data-wait-for-events for each set.-->
<div class="os-raise-ib-cta" data-button-text="Are you ready for more?" data-fire-event="RFM1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-cta-prompt">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<!--Start interaction. If multiple interactions appear under "are you ready for more?, they should all have matching "data-wait-for-event", which should also match the "data-fire-event" for the button. Note in this sample, they are all RFM1-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="69c0cbcd-2401-41e2-8c4f-6d0880fbf10e" data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<h4>Extending Your Thinking</h4>
<p>What quadratic equation of the form \(ax^2+bx+c=0\) would have the solutions \(x=5− \sqrt{2}\) and \(x=5+ \sqrt{2}\)?</p>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter the quadratic equation.</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(x^2−10x+23=0\). Working backward, these are solutions to \((x−5)^2=2\). In standard form, this is \(x^2−10x+23=0\).</p>
</div>
</div>
<!--End interaction-->
<br>
<!--Start interaction. If click to reveal is to appear under "are you ready for more?, they should all have matching "data-wait-for-event", which should also match the "data-fire-event" for the button. Note in this sample, they are all RFM1-->