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<!DOCTYPE html>
<html data-require="math math-format">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Solving quadratics by factoring 2</title>
<script src="../khan-exercise.js"></script>
<style type="text/css">
#answer_area input[type=text] {
width: 35px;
}
</style>
</head>
<body>
<div class="exercise">
<div class="vars">
<div data-ensure="abs(A) !== abs(B)">
<var id="A">randRangeNonZero( -10, 10 )</var>
<var id="B">randRangeNonZero( -10, 10 )</var>
</div>
<var id="SQUARE">randRange(1,6)</var>
</div>
<div class="problems">
<div id="original" data-weight="2">
<div class="vars">
<var id="CONSTANT">SQUARE*A*B</var>
<var id="SIMPLECONSTANT">A*B</var>
<var id="LINEAR">SQUARE*(-A-B)</var>
<var id="SIMPLELINEAR">-A-B</var>
</div>
<p class="problem">Determine where <code>f(x)</code> intersects the <code>x</code>-axis.</p>
<p class="question"><code>f(x) = <var>plus(SQUARE + "x^2")</var> + <var>plus( LINEAR + "x" )</var> + <var>CONSTANT</var></code></p>
<div class="solution" data-type="set">
<div class="set-sol"><var>A</var></div>
<div class="set-sol"><var>B</var></div>
<div class="input-format"><p style="font-size: 12px;">
<code>x = {}</code><span class="entry"></span><code>\space \text{and} \space x = {}</code><span class="entry"></span>
</p></div>
</div>
<div class="hints" data-apply="appendContents">
<div id="hint1">
<p>The two numbers <code class="hint_pink"><var>-A</var></code> and <code class="hint_pink"><var>-B</var></code> satisfy both conditions:</p>
<p><code>
\qquad \color{<var>PINK</var>}{<var>-A</var>} + \color{<var>PINK</var>}{<var>-B</var>} =
\color{<var>GREEN</var>}{<var>SIMPLELINEAR</var>}
</code></p>
<p><code>
\qquad \color{<var>PINK</var>}{<var>-A</var>} \times \color{<var>PINK</var>}{<var>-B</var>} =
\color{<var>BLUE</var>}{<var>SIMPLECONSTANT</var>}
</code></p>
</div>
<p id="hint2">
So <code>(x <var>A &lt; 0 ? "+" : ""</var> \color{<var>PINK</var>}{<var>-A</var>})
(x <var>B &lt; 0 ? "+" : ""</var> \color{<var>PINK</var>}{<var>-B</var>}) = 0</code>.
</p>
<p id="hint3">
Since <code>(x <var>A &lt; 0 ? "+" : ""</var> <var>-A</var>)
(x <var>B &lt; 0 ? "+" : ""</var> <var>-B</var>) = 0</code>,
we know that one or both quantities must equal zero for the equation to be true.
</p>
<p id="hint4"><code>x + <var>-A</var> = 0</code> or <code>x + <var>-B</var> = 0</code></p>
<p id="hint5"><b>Thus, <code>x = <var>A</var></code> and <code>x = <var>B</var></code> are the solutions.</b></p>
</div>
</div>
<div id="one-root" data-type="original" data-weight="1">
<div class="vars">
<var id="CONSTANT">SQUARE * A * A</var>
<var id="SIMPLECONSTANT">A * A</var>
<var id="LINEAR">SQUARE * ( -2 * A )</var>
<var id="SIMPLELINEAR">-2 * A</var>
</div>
<p class="problem">Determine where <code>f(x)</code> intersects the x-axis.</p>
<p class="question"><code>f(x) = <var>plus( SQUARE + "x^2")</var> + <var>plus( LINEAR + "x" )</var> + <var>CONSTANT</var></code></p>
<div class="solution" data-type="multiple">
<p><code>x = \quad</code><span class="sol"><var>A</var></span></p>
</div>
<div class="hints" data-apply="appendContents">
<div id="hint1">
<p>The number <code class="hint_pink"><var>-A</var></code> used twice satisfies both conditions:</p>
<p><code>
\qquad \color{<var>PINK</var>}{<var>-A</var>} + \color{<var>PINK</var>}{<var>-A</var>} =
\color{<var>GREEN</var>}{<var>SIMPLELINEAR</var>}
</code></p>
<p><code>
\qquad \color{<var>PINK</var>}{<var>-A</var>} \times \color{<var>PINK</var>}{<var>-A</var>} =
\color{<var>BLUE</var>}{<var>SIMPLECONSTANT</var>}
</code></p>
</div>
<p id="hint2">So <code>(x <var>A &lt; 0 ? "+" : ""</var> \color{<var>PINK</var>}{<var>-A</var>})^2 = 0</code>.</p>
<p id="hint3"><code>x + <var>-A</var> = 0</code></p>
<p id="hint4"><b>Thus, <code>x = <var>A</var></code> is the solution.</b></p>
</div>
</div>
</div>
<div class="hints">
<div data-if="SQUARE === 1">
<p>The function intersects the <code>x</code>-axis where <code>f(x) = 0</code>, so solve the equation:</p>
<p><code>
\qquad
<var>plus( SQUARE + "x^2" )</var>
<var>LINEAR &gt;= 0 ? "+" : ""</var>
<var>plus( "\\color{" + GREEN + "}{" + LINEAR + "}x" )</var>
<var>CONSTANT &gt;= 0 ? "+" : ""</var>
<var>plus( "\\color{" + BLUE + "}{" + CONSTANT + "}" )</var>
= 0
</code></p>
</div>
<div data-else>
<p>The function intersects the <code>x</code>-axis where <code>f(x) = 0</code>, so solve the equation:</p>
<p><code>\qquad <var>plus( SQUARE + "x^2", LINEAR + "x", CONSTANT )</var> = 0</code></p>
</div>
<div data-if="SQUARE > 1">
<p>Dividing both sides by <code><var>SQUARE</var></code> gives:</p>
<p><code>
\qquad x^2
<var>SIMPLELINEAR &gt;= 0 ? "+" : ""</var>
<var>plus( "\\color{" + GREEN + "}{" + SIMPLELINEAR + "}x" )</var>
<var>SIMPLECONSTANT &gt;= 0 ? "+" : ""</var>
<var>plus( "\\color{" + BLUE + "}{" + SIMPLECONSTANT + "}" )</var>
= 0
</code></p>
</div>
<p>
The coefficient on the <code>x</code> term is <code class="hint_green"><var>SIMPLELINEAR</var></code>
and the constant term is <code class="hint_blue"><var>SIMPLECONSTANT</var></code>, so we need to find two numbers
that <span class="hint_green">add up to <code><var>SIMPLELINEAR</var></code></span> and <span class="hint_blue">multiply to
<code><var>SIMPLECONSTANT</var></code></span>.
</p>
</div>
</div>
</body>
</html>
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