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<h4>Activity (15 minutes)</h4>
<p>By now, students recognize that when a quadratic equation is in the form of <em>expression</em> \(= 0\) and the
expression is in factored form, the equation can be solved using the zero product property. In this activity, they
encounter equations in which one side of the equal sign is not 0. To make one side equal 0 requires rearrangement. For
example, to solve \(x(x+6)=8\), the equation needs to be rearranged to \(x(x+6)−8=0\). Yet because the
expression on the left is no longer in factored form, the zero product property won't help after all, and another
strategy is needed.</p>
<p>Students recall that to solve a quadratic equation in the form of <em>expression</em> \(= 0\) is essentially to find
the zeros of a quadratic function defined by that expression, and that the zeros of a function correspond to the
horizontal intercepts of its graph. In the case of \(x(x+6)−8=0\), the function whose zeros we want to find is
defined by \(x(x+6)−8\). Graphing \(y=x(x+6)−8\) and examining the \(x\)-intercepts of the graph allows us
to see the number of solutions and what they are.</p>
<h4>Launch</h4>
<p>Arrange students in groups of two and provide access to graphing technology. Give students a moment to think quietly
about the first question and then ask them to briefly discuss their response with their partner before continuing with
the rest of the activity.</p>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
<p class="os-raise-extrasupport-name">Action and Expression: Provide Access for Physical Action</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Support effective and efficient use of tools and assistive technologies. To use graphing technology, some students may benefit from a demonstration or access to step-by-step instructions. </p>
<p class="os-raise-text-italicize">Supports accessibility for: Organization; Memory; Attention</p>
</div>
</div>
<br>
<h4>Student Activity </h4>
<p>Julio is solving three equations by graphing.</p>
<p>\((x−5)(x−3)=0\)</p>
<p>\((x−5)(x−3)=-1\)</p>
<p>\((x−5)(x−3)=-4\)</p>
<p>To solve the first equation, \((x−5)(x−3)=0\), he graphed \(y=(x−5)(x−3)\) and then looked
for the \(x\)-intercepts of the graph.</p>
<ol class="os-raise-noindent">
<li>Explain why the \(x\)-intercepts can be used to solve \((x−5)(x−3)=0\). </li>
</ol>
<p><strong>Answer:</strong> The solutions to \((x−5)(x−3)=0\) are \(x\)-values that make the expression have
a value of 0. At the \(x\)-intercepts, the \(y\)-value of the expression \((x−5)(x−3)\) is 0, so the
\(x\)-intercepts give the solutions.</p>
<ol class="os-raise-noindent" start="2">
<li>Graph \(y=(x−5)(x−3)\). What are the solutions?</li>
</ol>
<p>(Students were provided access to Desmos.)</p>
<p>Select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-text-bold">Answer:</p>
<p><img
alt="ALT TEXT: THE PARABOLA OPENS UP AND HAS A VERTEX AT THE POINT (4, NEGATIVE 1). THE \(x\)-intercepts ARE (3, 0) AND (5, 0)."
height="246" src="https://k12.openstax.org/contents/raise/resources/96200b7342d318aa4c33316b2b71f758009864f6"
width="562"></p>
<p>The solutions are \(x=5\) and \(x=3\).</p>
<p>To solve the second equation, Julio rewrote it as \((x−5)(x−3)+1=0\). He then graphed
\(y=(x−5)(x−3)+1\). </p>
<ol class="os-raise-noindent" start="3">
<li>Graph \(y=(x−5)(x−3)+1\). Then, use the graph to solve the equation. Be prepared to explain how you
used the graph for solving.</li>
</ol>
<p>(Students were provided access to Desmos.)</p>
<p>Select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-text-bold">Answer: </p>
<p><img alt="THE PARABOLA OPENS UP AND HAS A VERTEX AT THE POINT (4, 0). THE \(x\)-intercepts IS (4, 0)." height="285"
src="https://k12.openstax.org/contents/raise/resources/f3488bc6c2946bfe994975f99fd623c84876a3d3" width="624"></p>
<p>The solution is 4 for \((x−5)(x−3)+1=0\). The solution is the \(x\)-coordinate of the \(x\)-intercept.
</p>
<p>Solve the third equation using Julio's strategy of graphing the equations. </p>
<ol class="os-raise-noindent" start="4">
<li>What are the solutions to \(y=(x-5)(x-3)+4\)?</li>
</ol>
<p>(Students were provided access to Desmos.)</p>
<p>Select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-text-bold">Answer:</p>
<p><img alt="THE PARABOLA OPENS UP AND HAS A VERTEX AT THE POINT (4, 3). THE \(x\)-intercepts IS (4, 3)." height="284"
src="https://k12.openstax.org/contents/raise/resources/14e9433560eb5fcf0427971787cf1245ab48c391" width="624"></p>
<p>There are no solutions for \((x−5)(x−3)+4=0\). The graph has no \(x\)-intercept.</p>
<p>Think about the graphing strategy used to solve the equations and the solutions you found. View the Desmos graphs of
all three equations.
<p><img
src="https://k12.openstax.org/contents/raise/resources/242363d7056c9b9f08da49ce31c4dd77cb1400ee" alt="View of Desmos Graph of all three equations" width="483"></p>
<ol class="os-raise-noindent" start="5">
<li>Why might it be helpful to rearrange each equation to equal 0 on one side and then graph the expression on the
non-zero side?</li>
</ol>
<p><strong>Answer:</strong> When a quadratic expression is equal to 0 and we want to find the unknown values, we can
think of it as finding the zeros of a quadratic function, or finding the \(x\)-intercepts of the graph of that
function.</p>
<ol class="os-raise-noindent" start="6">
<li>How many solutions does each of the three equations have?</li>
</ol>
<p><strong>Answer:</strong> There are two solutions for the first equation, one solution for the second equation, and no
solution for the last equation.</p>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<ol class="os-raise-noindent">
<li>Graph \(y=(x−3)(x−5)\) and \(y=-1\) on the same coordinate plane. </li>
</ol>
<p>(Students were provided access to Desmos.)</p>
<p>How do the equations \(y=(x−3)(x−5)\) and \(y=-1\) help solve the equation \((x−3)(x−5)=-1\)?
</p>
<p><strong>Answer:</strong> The solution to \((x−3)(x−5)=-1\) is the \(x\)-value of the intersection point
of the two graphs of \(y=(x−3)(x−5)\) and \(y=-1\). This solution is 4.</p>
<p><img
alt="GRAPH OF A PARABOLA THAT OPENS UP WITH A VERTEX AT (4, NEGATIVE 1) AND \(x\)-intercepts OF 3 AND 5. A HORIZONTAL DOTTED LINE IS GRAPHED AT Y EQUALS NEGATIVE 1."
height="298" src="https://k12.openstax.org/contents/raise/resources/c9353744ffccea750a2f5c39059a17ddd6843fe1"
width="305"></p>
<ol class="os-raise-noindent" start="2">
<li>Graph \(y=(x−3)(x−5)\) and \(y=0\) on the same coordinate plane.</li>
</ol>
<p>(Students were provided access to Desmos.)</p>
<p>How do the equations \(y=(x−3)(x−5)\) and \(y=0\) help solve the equation \((x−3)(x−5)=0\)?
</p>
<p><strong>Answer:</strong> The solutions to \((x−3)(x−5)=0\) are the \(x\)-values of the intersection
points of the two graphs of \(y=(x−3)(x−5)\) and \(y=0\). These solutions are 3 and 5.</p>
<p><img
alt="GRAPH OF A PARABOLA THAT OPENS UP WITH A VERTEX AT (4, NEGATIVE 1) AND \(x\)-intercepts OF 3 AND 5. A HORIZONTAL DOTTED LINE IS GRAPHED AT Y EQUALS 0."
height="331" src="https://k12.openstax.org/contents/raise/resources/6bd98ec004ad28f418a6fb7a40e97ab780883f74"
width="337"></p>
<ol class="os-raise-noindent" start="3">
<li>Graph \(y = (x−3)(x−5)\) and \(y=3\) on the same coordinate plane.</li>
</ol>
<p>(Students were provided access to Desmos.)</p>
<p>How do the equations \(y = (x−3)(x−5)\) and \(y=3\) help solve the equation \((x−3)(x−5)=3\)?
</p>
<p><strong>Answer:</strong> The solutions to \((x−3)(x−5)=3\) are the \(x\)-values of the intersection
points of the two graphs of \(y=(x−3)(x−5)\) and \(y=3\). These solutions are 2 and 6.</p>
<p><img
alt="GRAPH OF A PARABOLA THAT OPENS UP WITH A VERTEX AT (4, NEGATIVE 1) AND \(x\)-intercepts OF 3 AND 5. A HORIZONTAL DOTTED LINE IS GRAPHED AT Y EQUALS 3."
height="344" src="https://k12.openstax.org/contents/raise/resources/7f2f82d8b8f1b91c55dcd84b20ad2ae04fa4e80f"
width="351"></p>
<ol class="os-raise-noindent" start="4">
<li>Use the graphs you created in questions 1–3 to help you find a few other equations of the form
\((x−3)(x−5)=z\) that have whole-number solutions.</li>
</ol>
<p><strong>Answer:</strong> There are many different answers. For example:</p>
<p>\((x−3)(x−5)=8\), \((x−3)(x−5)=15\), \((x−3)(x−5)=24\)</p>
<ol class="os-raise-noindent" start="5">
<li>Some of the values of \(z\) that you might have found are –1, 0, 3, 8, 15, 24, etc. Find a pattern in the
values of \(z\) that give whole-number solutions.</li>
</ol>
<p><strong>Answer:</strong> Starting at \(x=-1\), add 1, then 3, then 5, and so on, to find the next \(x\)-value that
yields whole-number solutions.</p>
<p>Because the expressions \(x−3\) and \(x−5\) represent numbers whose difference is 2, the values of \(z\)
have the form \(-1 \cdot 1\), \(0 \cdot 2\), \(1 \cdot 3\), \(2 \cdot 4\), and so on.</p>
<ol class="os-raise-noindent" start="6">
<li>Without solving, determine if \((x−5)(x−3)=120\) and \((x−5)(x−3)=399\) have whole-number
solutions. Be prepared to show your reasoning.</li>
</ol>
<p><strong>Answer:</strong> Both have whole-number solutions. The values of \(x\) that produce whole-number solutions
are all 1 less than a square number.</p>
<h4>Anticipated Misconceptions</h4>
<p>If students enter the equation \((x−5)(x−3)=0\) into their graphing technology, they may see an error
message, or they may see vertical lines. The lines will intersect the \(x\)-axis at the solutions, but they are
clearly not graphs of a quadratic function. Emphasize that we want to graph the function defined by
\(y=(x−5)(x−3)\) and use its \(x\)-intercepts to find the solution to the related equation. All the points
on the two vertical lines do represent solutions to the equation (because the points along each vertical line satisfy
the equation regardless of the value chosen for \(y\)), but understanding this is beyond the expectations for students
in this course.</p>
<h4>Activity Synthesis</h4>
<p>Invite students to share their responses, graphs, and explanations on how they used the graphs to solve the
equations. Discuss questions such as:</p>
<ul class="os-raise-noindent">
<li>"Are the original equation \((x−5)(x−3)=-1\) and the rewritten one \((x−5)(x−3)+1=0\)
equivalent?" (Yes, the equations are equivalent. In that example, 1 is added to both sides of the original
equation.) </li>
<li>"Why might it be helpful to rearrange the equation so that one side is an expression and the other side is 0?" (It
allows us to find the zeros of the function defined by that expression. The zeros correspond to the \(x\)-intercepts
of the graph.) </li>
<li>"What equation would you graph to solve this equation: \((x−4)(x−6)=15\)?"
(\((y=(x−4)(x−6)−15)\) "What about \((x+3)^2−1=5\)?" \((y=(x+3)^2−6)\) </li>
</ul>
<p>Make sure students understand that some quadratic functions have two zeros, some have one zero, and some have no
zeros, so their respective graphs will have two, one, or no horizontal intercepts, respectively.</p>
<p>Likewise, some quadratic equations have two solutions, some have one solution, and some have no real solutions.
(Because students won't know about numbers that aren't real until a future course, for now it is sufficient to say "no
solutions.")</p>
<h3>8.5.2: Self Check</h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the
concepts explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Use graphing technology to determine the number of solutions for the equation \((x+3)(x−6)=−22\).
Then find any solutions.</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
There is one solution. \(x=5.5\)
</td>
<td>
Incorrect. Let's try again a different way: Look again at the graph of \(y=(x+3)(x−6)+22\). There are
no \(x\)-intercepts. The answer is that there are no solutions.
</td>
</tr>
<tr>
<td>
There are two solutions. \(x=-6\) or \(x=3\)
</td>
<td>
Incorrect. Let's try again a different way: Look again at the graph of \(y=(x+3)(x−6)+22\). There are
no \(x\)-intercepts. The answer is that there are no solutions.
</td>
</tr>
<tr>
<td>
There are two solutions. \(x=-3\) or \(x=6\)
</td>
<td>
<p>Incorrect. Let's try again a different way: Look again at the graph of \(y=(x+3)(x−6)+22\). There are
no \(x\)-intercepts. The answer is that there are no solutions.
</td>
</tr>
<tr>
<td>
The equation has no solutions.
</td>
<td>
That's correct! Check yourself: Look again at the graph of \(y=(x+3)(x−6)+22\). There are no
\(x\)-intercepts. This means there are no solutions.
</td>
</tr>
</tbody>
</table>
<br>
<h3>8.5.2: Additional Resources</h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on their experience with
the self check. Students will not automatically have access to this content, so you may wish to share it with
those who could benefit from it. </em></p>
<h4>Interpreting Graphs to Solve Quadratic Equations</h4>
<p>Let's analyze and interpret the graphs of different quadratic equations.</p>
<p class="os-raise-text-bold">Example 1</p>
<p>First, let's look at the equation \((x−4)(x−7)=0\). </p>
<p>We will graph the equation \(y=(x−4)(x−7)\).</p>
<p><img alt="GRAPH OF A PARABOLA THAT OPENS UP AND PASSES THROUGH THE POINTS (4, 0) AND (7, 0)." height="491"
src="https://k12.openstax.org/contents/raise/resources/0f5ed86da7a118a883c063923369990eb0533368" width="491"></p>
<p>This graph has two \(x\)-intercepts.</p>
<p>The solutions to \((x−4)(x−7)=0\) are \(x\)-values that make the expression have a value of 0. </p>
<p>At the \(x\)-intercepts, the \(y\)-value of the expression \((x−4)(x−7)\) is 0, so the \(x\)-intercepts
give the solutions.</p>
<p>The \(x\)-intercepts and solutions to the equation are 4 and 7.</p>
<p>Note that the values of 4 and 7 could be referred to as solutions of the equation, \(x\)-intercepts of the graph,
roots of the equation, or zeros of the equation.</p>
<p class="os-raise-text-bold">Example 2</p>
<p>Let's look at a different example. </p>
<p>\((x−4)(x−7)=−2.25\) </p>
<p>First, we rewrite the equation to set it equal to 0. </p>
<p>\((x−4)(x−7)+2.25=0\)</p>
<p>We will graph the equation \(y=(x−4)(x−7)+2.25\).</p>
<p><img alt="GRAPH OF A PARABOLA THAT OPENS UP WITH A VERTEX AT (5.5, 0)." height="461"
src="https://k12.openstax.org/contents/raise/resources/1ef50a1728e0a8b372ea2439ea5d09509bbc8ad0" width="461"></p>
<p>This graph has one \(x\)-intercept. It is an example of a double root at 5.5. A double root is a root that appears
twice in the solution of an algebraic equation.</p>
<p>The solution is the \(x\)-coordinate of the \(x\)-intercept.</p>
<p>So, the solution to \((x−4)(x−7)=−2.25 is 5.5\).</p>
<p class="os-raise-text-bold">Example 3</p>
<p>Let's look at one last example. </p>
<p>\((x−4)(x−7)=-5\)</p>
<p>First, we rewrite the equation to set it equal to 0. </p>
<p>\((x−4)(x−7)+5=0\)</p>
<p>We will graph the equation \(y=(x−4)(x−7)+5\).</p>
<p><img
alt="GRAPH OF A PARABOLA THAT OPENS UP. THE GRAPH DOES NOT TOUCH OR CROSS THE X-AXIS. THE GRAPH PASSES THROUGH THE POINTS (4, 5) AND (7, 5)."
height="492" src="https://k12.openstax.org/contents/raise/resources/e1476e86da83ad0dfeb849b8243cc69255d2cb33"
width="492"></p>
<p>The graph has no \(x\)-intercepts.</p>
<p>So, there are no real solutions for \((x−4)(x−7)+5=0\).</p>
<p>Remember, when a quadratic expression is equal to 0 and we want to find the unknown values, we can think of it as
finding the zeros of a quadratic function, or finding the \(x\)-intercepts of the graph of that function.</p>
<h4>Try It: Interpreting Graphs to Solve Quadratic Equations</h4>
<p>Determine the number of solutions for the equation \((x−4)(x−9)=−6.25\). Then find any solutions.
</p>
<p>(Students were provided access to Desmos.)</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<h5> Solution</h5>
<p>Here is how to analyze the graph of the equation to find any solutions:</p>
<p><strong>Step 1</strong> - Rewrite the equation to set it equal to 0. </p>
<p> \((x−4)(x−9)+6.25=0\) </p>
<p><strong>Step 2</strong> - Graph the equation \(y=(x−4)(x−9)+6.25\). </p>
<p><img alt="GRAPH OF A PARABOLA THAT OPENS UP WITH A VERTEX AT THE POINT (6.5, 0)." height="483"
src="https://k12.openstax.org/contents/raise/resources/880a2a381249f44feb8864f1a3d0e1de69bdf077" width="483"></p>
<p><strong>Step 3</strong> - Identify the \(x\)-intercepts from the graph. </p>
<p> There is one \(x\)-intercept at \((6.5, 0)\)</p>
<p> This means there is one solution. </p>
<p> The solution is 6.5. </p>