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extraneous_solutions.html
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/
extraneous_solutions.html
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<!DOCTYPE html>
<html data-require="math math-format rational-expressions word-problems"><head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Extraneous solutions</title>
<script src="../khan-exercise.js"></script>
<style type="text/css">
#solutionarea.sol {
padding-left: 4px;
padding-right: 4px;
}
</style>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="A">randRangeNonZero(-10, 10)</var>
<var id="B">randRangeNonZero(-10, 10)</var>
<var id="NUMERATOR" data-ensure="NUMERATOR !== A || NUMERATOR !== B">randFromArray([A, B, randRangeNonZero(-10, 10)])</var>
<var id="CONSTANT">A * B</var>
<var id="COEFFICIENT">-A - B</var>
<var id="TERMS">(function() {
var coefficient1 = 0;
var coefficient2 = COEFFICIENT;
var constant1 = 0;
var constant2 = CONSTANT;
var simplify;
if (rand(2) < 1) {
coefficient1 = randRange(-COEFFICIENT, COEFFICIENT);
coefficient2 = COEFFICIENT - coefficient1;
simplify = coefficient1 !== 0 && coefficient2 !== 0;
} else {
constant1 = randRange(-CONSTANT, CONSTANT);
constant2 = CONSTANT - constant1;
simplify = constant1 !== 0 && constant2 !== 0;
}
return {
coefficient1: coefficient1,
coefficient2: coefficient2,
constant1: constant1,
constant2: constant2,
simplify: simplify
}
})()</var>
<var id="TERM1">new RationalExpression([[1, {x: 2}], [TERMS.coefficient1, 'x'], TERMS.constant1])</var>
<var id="TERM2">new RationalExpression([[-TERMS.coefficient2, 'x'], -TERMS.constant2])</var>
<var id="TERM2NEG">TERM2.multiply(-1)</var>
</div>
<div class="problems">
<div id="ignore-extraneous">
<p class="problem">Solve for <code>x</code>, ignoring any extraneous solutions:</p>
<p class="question">
<code>\dfrac{<var>TERM1</var>}{x - <var>NUMERATOR</var>}
= \dfrac{<var>TERM2</var>}{x - <var>NUMERATOR</var>}</code>
</p>
<div class="solution" data-type="set">
<span class="set-sol" data-type="number" data-if="NUMERATOR !== A"><var>A</var></span>
<span class="set-sol" data-type="number" data-if="NUMERATOR !== B && A !== B"><var>B</var></span>
<div class="input-format">
<div><code>x = </code> <span class="entry short50" data-fallback="0">a</span></div>
<div><code>x = </code> <span class="entry short50" data-fallback="0">a</span></div>
</div>
</div>
<div class="hints" data-apply="appendContents">
<div data-if="NUMERATOR === A || NUMERATOR === B">
<div>However, the original expression is undefined when <code>x = <var>NUMERATOR</var></code>.</div>
<div data-if="NUMERATOR !== A">Therefore, the only solution is <code>x = <var>A</var></code>.</div>
<div data-if="NUMERATOR !== B">Therefore, the only solution is <code>x = <var>B</var></code>.</div>
</div>
</div>
</div>
<div id="extraneous-exists">
<div class="vars">
<var id="ANSWER">NUMERATOR === A || NUMERATOR === B ? "<code>x = " + NUMERATOR + "</code>" : "No extraneous solutions"</var>
<var id="POSSIBILITIES">shuffle([
"<code>x = " + NUMERATOR + "</code>",
"<code>x = " + (-NUMERATOR) + "</code>",
])</var>
</div>
<p class="problem">What is the extraneous solution to these equations?</p>
<p class="question">
<code>\dfrac{<var>TERM1</var>}{x - <var>NUMERATOR</var>}
= \dfrac{<var>TERM2</var>}{x - <var>NUMERATOR</var>}</code>
</p>
<div class="solution"><var>ANSWER</var></div>
<ul class="choices" data-category="true">
<li><var>POSSIBILITIES[0]</var></li>
<li><var>POSSIBILITIES[1]</var></li>
<li>No extraneous solutions</li>
</ul>
<div class="hints" data-apply="appendContents">
<div data-if="NUMERATOR === A || NUMERATOR === B" data-unwrap="">
<p>At <code>x = <var>NUMERATOR</var></code>, the denominator of the original expression is 0.</p>
<p>Since the expression is undefined at <code>x = <var>NUMERATOR</var></code>, it is an extraneous solution.</p>
</div>
<div data-else="">The original expression is defined at <code>x = <var>A</var></code> and <code>x = <var>B</var></code>, so there are no extraneous solutions.</div>
</div>
</div>
</div>
<div class="hints">
<div>
<p>Multiply both sides by <code>x - <var>NUMERATOR</var></code>:</p>
<p><code>\qquad \dfrac{<var>TERM1</var>}{x - <var>NUMERATOR</var>} (x - <var>NUMERATOR</var>)
= \dfrac{<var>TERM2</var>}{x - <var>NUMERATOR</var>} (x - <var>NUMERATOR</var>)</code></p>
<p><code>\qquad <var>TERM1</var> = <var>TERM2</var></code></p>
</div>
<div>
<p>Subtract <code><var>TERM2</var></code> from both sides:</p>
<p><code>\qquad <var>TERM1</var> - (<var>TERM2</var>) = <var>TERM2</var> - (<var>TERM2</var>)</code></p>
<p><code>\qquad <var>TERM1</var> + <var>TERM2NEG</var> = 0</code></p>
<p data-if="TERMS.simplify"><code>\qquad <var>TERM1.add(TERM2NEG)</var> = 0</code></p>
</div>
<div>
<p>Factor the expression:</p>
<p><code>\qquad (x - <var>A</var>)(x - <var>B</var>) = 0</code></p>
</div>
<p>Therefore <code>x = <var>A</var></code> <span data-if="A !== B">or <code>x = <var>B</var></code></span></p>
</div>
</div>
</body></html>