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79408cc1-1530-404f-b70e-64b64d02a82d.html
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<p><strong><em>Students will complete the following questions to practice the skills they have learned in this lesson.</em></strong></p>
<ol class="os-raise-noindent">
<li>The quadratic equation \(x^2+7x+10=0\) is in the form of \(ax^2+bx+c=0\).</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>What is the value of \(a\)?</li>
</ol>
<p><strong>Answer:</strong> 1</p>
<ol class="os-raise-noindent" start="2" type="a">
<li>What is the value of \(b\)?</li>
</ol>
<p><strong>Answer:</strong> 7</p>
<ol class="os-raise-noindent" start="3" type="a">
<li>What is the value of \(c\)?</li>
</ol>
<p><strong>Answer:</strong> 10</p>
<p>Examine the solution method that has been started below and use it to answer parts d and e.</p>
<p>Original equation<br>
\(x^2+7x+10=0\)</p>
<p><strong>Step 1</strong> - Subtract 10 from each side.<br>
\(x^2+7x=-10\)</p>
<p><strong>Step 2</strong> - Multiply each side by 4.<br>
\(4x^2+4(7x)=-4(10)\)</p>
<p><strong>Step 3</strong> - Rewrite \(4x^2\) as \((2x)^2\) and \(4(7x)\) as \(2(7)2x\).<br>
\((2x)^2+2(7)(2x) +\_\_\_\_^2=\_\_\_\_^2-4(10)\)</p>
<p><strong>Step 4</strong> - \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)<br>
\((2x+\_\_\_\_)^2=\_\_\_\_^2-4(10)\)</p>
<p><strong>Step 5</strong> - \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)<br>
\(2x+\_\_\_\_= \pm \sqrt{\_\_\_\_^2-4(10)}\)</p>
<p><strong>Step 6</strong> - \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)<br>
\(2x=\_\_\_\_\pm \sqrt{\_\_\_\_^2-4(10)}\)</p>
<p><strong>Step 7</strong> - \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)<br>
\(x= \_\_\_\_\)</p>
<ol class="os-raise-noindent" start="4" type="a">
<li>In Step 2, what might be a good reason for multiplying each side of the equation by 4?</li>
</ol>
<ul class="os-raise-noindent">
<li> You must always multiply both sides by 4 when completing the square. </li>
<li> If the value of \(c\) is positive and you multiply by 4, it will be easier to divide the \(b\) term by 3 before you square it. </li>
<li> Multiplying by 4 makes the coefficient of the \(b\) term a perfect square, which makes it easier to complete the square. </li>
<li> Multiplying by 4 makes the coefficient of the squared term a perfect square, which makes it easier to complete the square. </li>
</ul>
<p><strong>Answer:</strong> Multiplying by 4 makes the coefficient of the squared term a perfect square, which makes it easier to complete the square.</p>
<ol class="os-raise-noindent" start="5" type="a">
<li>Complete the unfinished steps, then <strong>select two</strong> solutions.</li>
</ol>
<ul class="os-raise-noindent">
<li> -10 </li>
<li> -5 </li>
<li> -2 </li>
<li> -4 </li>
</ul>
<p><strong>Answer:</strong> The solutions are -5 and -2.</p>
<p>Here are the unfinished steps:</p>
<p><strong>Step 3</strong> - Rewrite \(4x^2\) as \((2x)^2\) and \(4(7x)\) as \(2(7)2x\).<br>
\((2x)^2+2(7)(2x) +(7)^2=(7)^2-4(10)\)</p>
<p><strong>Step 4</strong> - Factor the trinomial.<br>
\((2x+7)^2=(7)^2-4(10)\)</p>
<p><strong>Step 5</strong> - Take the square root of each side.<br>
\(2x+7= \pm \sqrt{7^2-4(10)}\)</p>
<p><strong>Step 6</strong> - Subtract 7 from both sides.<br>
\(2x=-7 \pm \sqrt{7^2-4(10)}\)</p>
<p><strong>Step 7</strong> - Solve.<br>
\(x=-5\), -2</p>
<ol class="os-raise-noindent" start="6" type="a">
<li>Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula, \(x=\frac{-b \pm \sqrt{b^2- 4ac}}{2a}\), but do not evaluate any of the expressions. Explain how this expression is related to solving \(x^2+7x+10=0\) by completing the square.</li>
</ol>
<ul class="os-raise-noindent">
<li> In both forms of solving the equation, you need to list the values of \(a\), \(b\), and \(c\) to determine the solutions. </li>
<li> Rather than evaluating at each step, the calculation is done all at once, at the end. </li>
<li> They are related because neither one can have a negative \(c\) value. </li>
<li> For this equation, the quadratic formula and completing the square are not the same and therefore not related. </li>
</ul>
<p><strong>Answer:</strong> Rather than evaluating at each step, the calculation is done all at once, at the end.</p>
<ol class="os-raise-noindent" start="2">
<li>Consider the standard form of the equation \(x^2−39=0\).</li>
</ol>
<ol class="os-raise-noindent" type="a">
<li>What is the value of \(a\)?</li>
</ol>
<p><strong>Answer:</strong> 1</p>
<ol class="os-raise-noindent" start="2" type="a">
<li>What is the value of \(b\)?</li>
</ol>
<p><strong>Answer:</strong> 0</p>
<ol class="os-raise-noindent" start="3" type="a">
<li>What is the value of \(c\)?</li>
</ol>
<p><strong>Answer:</strong> -39</p>
<ol class="os-raise-noindent" start="4" type="a">
<li>Can you use the quadratic formula to solve this equation?</li>
</ol>
<ul class="os-raise-noindent">
<li> Yes </li>
<li> No </li>
</ul>
<p><strong>Answer:</strong> Yes<br>
The quadratic formula works for all quadratic equations. Substituting the values of \(a\), \(b\), and \(c\) into the quadratic formula and evaluating the expression gives the solutions.</p>
<ol class="os-raise-noindent" start="5" type="a">
<li>Can you solve this equation using square roots?</li>
</ol>
<ul class="os-raise-noindent">
<li> Yes, the solutions are \(\pm 39\). </li>
<li> No, there is no solution. </li>
<li> Yes, the solutions are \(\pm \sqrt{39}\). </li>
<li> No, this equation can only be solved by the quadratic formula. </li>
</ul>
<p><strong>Answer:</strong> Yes, the solutions are \(\pm \sqrt{39}\).<br>
Adding 39 to both sides of the equation gives \(x^2=39\). The solutions are \(\pm \sqrt{39}\).</p>
<ol class="os-raise-noindent" start="3">
<li>Clare is deriving the quadratic formula by solving \(ax^2+bx+c=0\) by completing the square.</li>
</ol>
<p>She arrived at this equation: \((2ax+b)^2=b^2−4ac\).</p>
<p>Choose the best description of what she needs to do to finish solving for \(x\).</p>
<ul class="os-raise-noindent">
<li> Subtract \(b^2\) from each side. Find the square roots of each side. Then divide each side by 2. </li>
<li> Find the square roots of each side. Subtract \(b \) from each side. Then divide each side by \(2a\). </li>
<li> Divide each side by 2. Subtract \(b \) from each side. Then find the square roots of each side. </li>
<li> Find the square roots of each side. Divide each side by 2. Subtract \(b \) from each side. </li>
</ul>
<p><strong>Answer:</strong> She needs to find the square roots of each side. Subtract \(b \) from each side. Then divide each side by \(2a\).</p>