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<!DOCTYPE html>
<html data-require="math math-format">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Solving quadratics by taking the square root</title>
<script src="../khan-exercise.js"></script>
<style type="text/css">
#answer_area input[type=text] {
width: 35px;
<div class="exercise">
<div class="problems">
<div id="solve">
<div class="vars">
<var id="H">randRangeNonZero( -10, 10 )</var>
<var id="CONSTANT">randRange( 1, 9 )</var>
<var id="K">-( CONSTANT * CONSTANT )</var>
<p class="problem">Solve for <code>x</code>:</p>
<p class="question"><code>(x - <var>H</var>)^2 + <var>K</var> = 0</code></p>
<div class="solution" data-type="set">
<div class="set-sol"><var>H + CONSTANT</var></div>
<div class="set-sol"><var>H - CONSTANT</var></div>
<div class="input-format"><p style="font-size: 12px;">
<code>x = {}</code><span class="entry"></span><code>\space \text{or} \space x = {}</code><span class="entry"></span>
<p class="example">integers, like <code>6</code></p>
<p class="example"><em>simplified proper</em> fractions, like <code>3/5</code></p>
<p class="example"><em>simplified improper</em> fractions, like <code>7/4</code></p>
<p class="example">and/or <em>exact</em> decimals, like <code>0.75</code></p>
<div class="hints">
<p>Add <code><var>abs( K )</var></code> to both sides so we can start isolating <code>x</code> on the left:</p>
<p><code>\qquad (x - <var>H</var>)^2 = <var>-K</var></code></p>
<p>Take the square root of both sides to get rid of the exponent.</p>
<p><code>\qquad \sqrt{(x - <var>H</var>)^2} = \pm \sqrt{<var>-K</var>}</code></p>
Be sure to consider both positive and negative <code><var>CONSTANT</var></code>, since squaring either one results
in <code><var>-K</var></code>.
<p><code>\qquad x - <var>H</var> = \pm <var>CONSTANT</var></code></p>
<span data-if="H > 0">Add <code><var>abs( H )</var></code> to both sides to isolate <code>x</code> on the left:</span>
<span data-else>Subtract <code><var>abs( H )</var></code> from both sides to isolate <code>x</code> on the left:</span>
<p><code>\qquad x = <var>H</var> \pm <var>CONSTANT</var></code></p>
<p class="final_answer">Add and subtract <code><var>CONSTANT</var></code> to find the two possible solutions:</p>
<p><code>\qquad x = <var>H + CONSTANT</var> \quad \text{or} \quad x = <var>H - CONSTANT</var></code></p>
<div id="intersect" data-type="solve">
<p class="problem">Determine where <code>f(x)</code> intersects the <code>x</code>-axis.</p>
<p class="question"><code>f(x) = (x - <var>H</var>)^2 + <var>K</var></code></p>
<div class="hints" data-apply="prependContents">
<p>The function intersects the <code>x</code>-axis where <code>f(x) = 0</code>, so solve the equation:</p>
<p><code>\qquad (x - <var>H</var>)^2 + <var>K</var> = 0</code></p>
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