e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
444 lines
20 KiB
HTML
444 lines
20 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/quadratic-equation.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:37:00 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Quadratic Equations</title>
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="preload" href="../images/style/font-champ-bold.ttf" as="font" type="font/ttf" crossorigin="">
|
||
<link rel="preload" href="../style4.css" as="style">
|
||
<link rel="preload" href="../main4.js" as="script">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
|
||
<h1 class="center">Quadratic Equations</h1>
|
||
|
||
<p>An example of a <b>Quadratic Equation</b>:</p>
|
||
<p class="center"><img src="images/quadratic-equation-reason.svg" alt="A Quadratic Equation 5x^2 - 3x + 3 = 0" height="63" width="296"></p>
|
||
<p>The function makes nice curves like this one:</p>
|
||
<p class="center"><a href="../geometry/parabola.html"><img src="images/quadratic-soccer.gif" alt="quadratic soccer kick" height="150" width="266"></a></p>
|
||
|
||
|
||
<h2>Name</h2>
|
||
|
||
<div class="words">
|
||
<p>The name <b>Quadratic</b> comes from "quad" meaning square, because the variable gets <a href="../square-root.html">squared</a> (like <b>x<sup>2</sup></b>).</p>
|
||
<p>It is also called an "Equation of <a href="degree-expression.html">Degree</a> 2" (because of the "2" on the <b>x</b>)</p>
|
||
</div>
|
||
|
||
|
||
<h2>Standard Form</h2>
|
||
|
||
<p>The <b>Standard Form</b> of a Quadratic Equation looks like this:</p>
|
||
<p class="center"><img src="images/quadratic-equation.svg" alt="Quadratic Equation: ax^2 + bx + c = 0" height="33" width="301"><br></p>
|
||
<ul>
|
||
<li><b>a</b>, <b>b</b> and <b>c</b> are known values. <b>a</b> can't be 0.</li>
|
||
</ul>
|
||
<ul>
|
||
<li>"<b>x</b>" is the <b><a href="definitions.html">variable</a></b> or unknown (we don't know it yet).</li>
|
||
</ul>
|
||
<p> </p>
|
||
<p>Here are some examples:</p>
|
||
<div class="simple">
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td class="larger" style="white-space: nowrap;" align="right" valign="top" nowrap="nowrap"><b>2x<sup>2</sup> + 5x + 3 = 0</b></td>
|
||
<td style="width:20px;"> </td>
|
||
<td>In this one <b>a=2</b>, <b>b=5</b> and <b>c=3</b></td>
|
||
</tr>
|
||
<tr>
|
||
<td align="right" valign="top" nowrap="nowrap"> </td>
|
||
<td> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td class="larger" align="right" height="91" valign="top" nowrap="nowrap"><b>x<sup>2</sup> − 3x = 0</b></td>
|
||
<td> </td>
|
||
<td height="91"> This one is a little more tricky:
|
||
|
||
|
||
|
||
|
||
<ul>
|
||
<li>Where is <b>a</b>? Well <b>a=1</b>, as we don't usually write "1x<sup>2</sup>"</li>
|
||
<li><b>b = −3</b></li>
|
||
<li>And where is <b>c</b>? Well <b>c=0</b>, so is not shown.</li>
|
||
</ul>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td class="larger" align="right" valign="top" nowrap="nowrap"><b>5x − 3 = 0</b></td>
|
||
<td> </td>
|
||
<td><b>Oops!</b> This one is <b>not</b> a quadratic equation: it is missing <b>x<sup>2</sup></b><br>
|
||
(in other words <b>a=0</b>, which means it can't be quadratic)</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p> </p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><a href="quadratic-equation-graph.html"><img src="images/quadratic-graph.svg" alt="Quadratic Graph" height="140" width="175"></a></p>
|
||
|
||
|
||
<h2>Have a Play With It</h2>
|
||
|
||
<p>Play with the "<a href="quadratic-equation-graph.html">Quadratic Equation Explorer</a>" so you can see:</p>
|
||
<ul>
|
||
<li>the function's graph, and</li>
|
||
<li>the solutions (called "roots").</li>
|
||
</ul>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Hidden Quadratic Equations!</h2>
|
||
|
||
<p>As we saw before, the <b>Standard Form</b> of a Quadratic Equation is</p>
|
||
<div class="def">
|
||
<p class="center larger">ax<sup>2</sup> + bx + c = 0</p>
|
||
</div>
|
||
<p>But sometimes a quadratic equation does not look like that!</p>
|
||
<p>For example:</p>
|
||
<div class="simple">
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<th>In disguise</th>
|
||
<th class="large"><img src="../images/style/right-arrow.gif" alt="right arrow" height="46" width="46"></th>
|
||
<th>In Standard Form</th>
|
||
<th>a, b and c</th>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td nowrap="nowrap"><b>x<sup>2</sup> = 3x − 1</b></td>
|
||
<td>Move all terms to left hand side</td>
|
||
<td nowrap="nowrap"><b>x<sup>2</sup> − 3x + 1 = 0</b></td>
|
||
<td>a=1, b=−3, c=1</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td nowrap="nowrap"><b>2(w<sup>2</sup> − 2w) = 5</b></td>
|
||
<td><a href="expanding.html">Expand</a> (undo the <a href="brackets.html">brackets</a>),<br>
|
||
and move 5 to left</td>
|
||
<td nowrap="nowrap"><b>2w<sup>2</sup> − 4w − 5 = 0</b></td>
|
||
<td>a=2, b=−4, c=−5</td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td nowrap="nowrap"><b>z(z−1) = 3</b></td>
|
||
<td>Expand, and move 3 to left</td>
|
||
<td nowrap="nowrap"><b>z<sup>2</sup> − z − 3 = 0</b></td>
|
||
<td>a=1, b=−1, c=−3</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p> </p>
|
||
|
||
|
||
<h2>How To Solve Them?</h2>
|
||
|
||
<div class="words">
|
||
<p>The "<b>solutions</b>" to the Quadratic Equation are where it is <b>equal to zero</b>.</p>
|
||
<p>They are also called "<b>roots</b>", or sometimes "<b>zeros</b>"</p>
|
||
</div>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/quadratic-graph.svg" alt="Quadratic Graph" height="140" width="175"></p>
|
||
<p> </p>
|
||
<p>There are usually 2 solutions (as shown in this graph).</p>
|
||
<p> </p>
|
||
<p>And there are a few different ways to find the solutions:</p>
|
||
<div style="clear:both"></div>
|
||
<div class="dotpoint"> We can <a href="factoring-quadratics.html">Factor the Quadratic</a> (find what to multiply to make the Quadratic Equation) </div>
|
||
<div class="dotpoint"> Or we can <a href="completing-square.html">Complete the Square</a></div>
|
||
<div class="dotpoint"> Or we can use the special <b>Quadratic Formula</b>:
|
||
|
||
|
||
|
||
|
||
<p class="center"><img src="images/quadratic-formula.svg" alt="Quadratic Formula: x = [ -b (+-) sqrt(b^2 - 4ac) ] / 2a" height="79" width="286"></p>
|
||
<p class="center">Just plug in the values of a, b and c, and do the calculations.</p>
|
||
<p>We will look at this method in more detail now.</p>
|
||
</div>
|
||
|
||
|
||
<h2>About the Quadratic Formula</h2>
|
||
|
||
<h3>Plus/Minus</h3>
|
||
<p>First of all what is that plus/minus thing that looks like <span class="largest">± </span>?</p>
|
||
<p class="so">The <span class="large">±</span> means there are TWO answers:</p>
|
||
<p class="center larger">x = <span class="intbl"> <em>−b <span class="hilite">+</span> √(b<sup>2 </sup>− 4ac)</em> <strong>2a</strong> </span></p>
|
||
<p class="center larger">x = <span class="intbl"> <em>−b <span class="hilite">−</span> √(b<sup>2 </sup>− 4ac)</em> <strong>2a</strong></span></p>
|
||
<p>Here is an example with two answers:</p>
|
||
<p class="center"><img src="images/quadratic-graph.svg" alt="Quadratic Graph" height="140" width="175"></p>
|
||
<p>But it does not always work out like that!</p>
|
||
<ul>
|
||
<li>Imagine if the curve "just touches" the x-axis.</li>
|
||
<li>Or imagine the curve is so <b>high</b> it doesn't even cross the x-axis!</li></ul>
|
||
<p>This is where the "Discriminant" helps us ...</p>
|
||
|
||
<h3>Discriminant</h3>
|
||
<p class="indent50px">Do you see <b>b<sup>2</sup> − 4ac</b> in the formula above? It is called the <b>Discriminant</b>, because it can "discriminate" between the possible types of answer:</p>
|
||
<div class="bigul">
|
||
<ul>
|
||
<li class="indent50px">when <b>b<sup>2</sup> − 4ac</b> is positive, we get two <a href="../numbers/real-numbers.html">Real</a> solutions</li>
|
||
<li class="indent50px">when it is zero we get just ONE real solution (both answers are the same)</li>
|
||
<li class="indent50px">when it is negative we get a pair of <a href="../numbers/complex-numbers.html">Complex</a> solutions</li>
|
||
</ul>
|
||
</div>
|
||
<p class="indent50px"><i>Complex solutions?</i> Let's talk about them after we see how to use the formula.</p>
|
||
<p> </p>
|
||
|
||
<h3>Using the Quadratic Formula</h3>
|
||
<p>Just put the values of a, b and c into the Quadratic Formula, and do the calculations.</p>
|
||
<div class="example">
|
||
|
||
<h3>Example: Solve 5x<sup>2</sup> + 6x + 1 = 0</h3>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Coefficients are:</span><span class="right">a = 5, b = 6, c = 1</span></div>
|
||
<div class="row"><span class="left">Quadratic Formula:</span><span class="right"><span class="center">x = <span class="intbl">
|
||
<em>−b ± √(b<sup>2 </sup>− 4ac)</em>
|
||
<strong>2a</strong>
|
||
</span></span>
|
||
</span></div>
|
||
<div class="row"><span class="left">Put in a, b and c:</span><span class="right"><span class="center">x = <span class="intbl">
|
||
<em>−6 ± √(6<sup>2 </sup>− 4×5×1)</em>
|
||
<strong>2×5</strong>
|
||
</span></span>
|
||
</span></div>
|
||
<div class="row"><span class="left">Solve:</span><span class="right"><span class="center">x = <span class="intbl">
|
||
<em>−6 ± √(36− 20)</em>
|
||
<strong>10</strong>
|
||
</span></span>
|
||
</span></div>
|
||
<div class="row"><span class="left"> </span><span class="right"><span class="center">x = <span class="intbl">
|
||
<em>−6 ± √(16)</em>
|
||
<strong>10</strong>
|
||
</span></span>
|
||
</span></div>
|
||
<div class="row"><span class="left"> </span><span class="right"><span class="center">x = <span class="intbl">
|
||
<em>−6 ± 4</em>
|
||
<strong>10</strong>
|
||
</span></span>
|
||
</span></div>
|
||
<div class="row"><span class="left"> </span><span class="right">x = −0.2 <b>or</b> −1</span></div>
|
||
</div>
|
||
<p> </p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/graph-5x2p6xp1.svg" alt="5x^2+6x+1" height="" width=""></p>
|
||
<p class="larger center"><b>Answer:</b> x = −0.2 <b>or</b> x = −1</p>
|
||
<p> </p>
|
||
<p>And we see them on this graph.</p>
|
||
<div style="clear:both"></div>
|
||
<table style="border: 0;">
|
||
<tbody>
|
||
<tr>
|
||
<td valign="top">Check <b>-0.2</b>:</td>
|
||
<td width="20" valign="top"> </td>
|
||
<td>5×(<b>−0.2</b>)<sup>2</sup> + 6×(<b>−0.2</b>) + 1<br>
|
||
= 5×(0.04) + 6×(−0.2) + 1<br>
|
||
= 0.2 − 1.2 + 1<br>
|
||
<b>= 0</b></td>
|
||
</tr>
|
||
<tr>
|
||
<td valign="top">Check <b>-1</b>:</td>
|
||
<td valign="top"> </td>
|
||
<td>5×(<b>−1</b>)<sup>2</sup> + 6×(<b>−1</b>) + 1<br>
|
||
= 5×(1) + 6×(−1) + 1<br>
|
||
= 5 − 6 + 1<br>
|
||
<b>= 0</b></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<h3>Remembering The Formula</h3>
|
||
<p>A kind reader suggested singing it to "Pop Goes the Weasel":</p>
|
||
<div class="center80">
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td rowspan="2"><span class="times">♫</span> </td>
|
||
<td><i><b>"x is equal to minus b</b></i></td>
|
||
<td class="times" width="40"> </td>
|
||
<td rowspan="2"><span class="times">♫</span> </td>
|
||
<td><i>"All around the mulberry bush</i></td>
|
||
</tr>
|
||
<tr>
|
||
<td><b><i>plus or minus the square root</i></b></td>
|
||
<td style="width:40px;"> </td>
|
||
<td><i>The monkey chased the weasel</i></td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td><b><i>of b-squared minus four a c</i></b></td>
|
||
<td style="width:40px;"> </td>
|
||
<td> </td>
|
||
<td><i>The monkey thought 'twas all in fun</i></td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td><b><i>ALL over two a"</i></b></td>
|
||
<td style="width:40px;"> </td>
|
||
<td> </td>
|
||
<td><i>Pop! goes the weasel"</i></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>Try singing it a few times and it will get stuck in your head!</p>
|
||
<p>Or you can remember this story:</p>
|
||
<div class="center80">
|
||
<p class="center">x = <span class="intbl">
|
||
<em>−b ± √(b<sup>2 </sup>− 4ac)</em>
|
||
<strong>2a</strong>
|
||
</span></p>
|
||
<p class="center"><i>"A negative boy was thinking yes or no about going to a party,<br>
|
||
at the party he talked to a square boy but not to the 4 awesome chicks.<br>
|
||
It was all over at 2 am.</i>"</p>
|
||
</div>
|
||
|
||
|
||
<h2>Complex Solutions?</h2>
|
||
|
||
<p>When the Discriminant (the value <b>b<sup>2</sup> − 4ac</b>) is negative we get a pair of <a href="../numbers/complex-numbers.html">Complex</a> solutions ... what does that mean?</p>
|
||
<p>It means our answer will include <a href="../numbers/imaginary-numbers.html">Imaginary Numbers</a>. Wow!</p>
|
||
<div class="example">
|
||
|
||
<h3>Example: Solve 5x<sup>2</sup> + 2x + 1 = 0</h3>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"><b>Coefficients</b> are<b>:</b></span><span class="right">a=5, b=2, c=1</span></div>
|
||
<div class="row"><span class="left">Note that the <b>Discriminant</b> is negative:</span><span class="right">b<sup>2</sup> − 4ac = 2<sup>2</sup> − 4×5×1<br>
|
||
= <b>−16</b></span></div>
|
||
<div class="row"><span class="left">Use the <b>Quadratic Formula:</b></span><span class="right"><span class="center">x = <span class="intbl">
|
||
<em>−2 ± √(−16)</em>
|
||
<strong>10</strong>
|
||
</span></span>
|
||
</span></div>
|
||
<p><em>√(−16)</em>
|
||
= 4<b>i</b><br>
|
||
(where <b>i</b> is the imaginary number √−1)</p>
|
||
<div class="row"><span class="left">So:</span><span class="right"><span class="center">x = <span class="intbl">
|
||
<em>−2 ± 4<b>i</b></em>
|
||
<strong>10</strong>
|
||
</span></span>
|
||
</span></div>
|
||
</div><p> </p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/graph-5x2p2xp1.gif" alt="5x^2+6x+1" height="181" width="178"></p>
|
||
<p class="larger center"><b>Answer:</b> x = −0.2 ± 0.4<b>i</b></p>
|
||
<p> </p>
|
||
<p>The graph does not cross the x-axis. That is why we ended up with complex numbers.</p>
|
||
<div style="clear:both"></div>
|
||
</div>
|
||
<p>In a way it is easier: we don't need more calculation, we leave it as <span class="larger">−0.2 ± 0.4<b>i</b></span>.</p>
|
||
<div class="example">
|
||
|
||
<h3>Example: Solve x<sup>2</sup> − 4x + 6.25 = 0</h3>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left"><b>Coefficients</b> are<b>:</b></span><span class="right">a=1, b=−4, c=6.25</span></div>
|
||
<div class="row"><span class="left">Note that the <b>Discriminant</b> is negative:</span><span class="right">b<sup>2</sup> − 4ac = (−4)<sup>2</sup> − 4×1×6.25<br>
|
||
= <b>−9</b></span></div>
|
||
<div class="row"><span class="left">Use the <b>Quadratic Formula:</b></span><span class="right"><span class="center">x = <span class="intbl"> <em>−(−4) ± √(−9)</em> <strong>2</strong> </span></span> </span></div>
|
||
<p><em>√(−9)</em> = 3<b>i</b><br>
|
||
(where <b>i</b> is the imaginary number √−1)</p>
|
||
<div class="row"><span class="left">So:</span><span class="right"><span class="center">x = <span class="intbl"> <em>4 ± 3<b>i</b></em> <strong>2</strong> </span></span> </span></div>
|
||
</div>
|
||
<p> </p>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/quadratic-graph-complex-a.svg" alt="Quadratic Graph with Cmplex Roots" height="140" width="175"></p>
|
||
<p class="larger center"><b>Answer:</b> x = 2 ± 1.5<b>i</b></p>
|
||
<p> </p>
|
||
<p>The graph does not cross the x-axis. That is why we ended up with complex numbers.</p>
|
||
<div style="clear:both"></div>
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/quadratic-graph-complex.svg" alt="Quadratic Graph with Cmplex Roots" height="140" width="175"></p>
|
||
<p>BUT an upside-down mirror image of our equation does cross the x-axis at <span class="larger center">2 ± 1.5</span> (note: missing the <b>i</b>).</p>
|
||
<p>Just an interesting fact for you!</p>
|
||
<p><br></p>
|
||
<p><br></p>
|
||
<div style="clear:both"></div>
|
||
</div>
|
||
|
||
<p> </p>
|
||
|
||
|
||
<h2>Summary</h2>
|
||
|
||
<ul class="larger">
|
||
<li>Quadratic Equation in Standard Form: ax<sup>2</sup> + bx + c = 0</li>
|
||
<li>Quadratic Equations can be <a href="factoring-quadratics.html">factored</a></li>
|
||
<li>Quadratic Formula: <span class="center">x = <span class="intbl">
|
||
<em>−b ± √(b<sup>2 </sup>− 4ac)</em>
|
||
<strong>2a</strong>
|
||
</span></span></li>
|
||
<li>When the Discriminant (<b>b<sup>2</sup>−4ac</b>) is:
|
||
|
||
|
||
|
||
|
||
<ul>
|
||
<li>positive, there are 2 real solutions</li>
|
||
<li>zero, there is one real solution</li>
|
||
<li>negative, there are 2 complex solutions</li>
|
||
</ul></li>
|
||
</ul><p> </p>
|
||
<div class="questions">360, 361, 1201, 1202, 2333, 2334, 3894, 3895, 2335, 2336</div>
|
||
|
||
<div class="related">
|
||
<a href="../quadratic-equation-solver.html">Quadratic Equation Solver</a>
|
||
<a href="factoring-quadratics.html">Factoring Quadratics</a>
|
||
<a href="completing-square.html">Completing the Square</a>
|
||
<a href="quadratic-equation-graphing.html">Graphing Quadratic Equations</a>
|
||
<a href="quadratic-equation-real-world.html">Real World Examples of Quadratic Equations</a>
|
||
<a href="quadratic-equation-derivation.html">Derivation of Quadratic Equation</a>
|
||
<a href="index.html">Algebra Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2021 MathsIsFun.com</div>
|
||
|
||
</div>
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/quadratic-equation.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:37:02 GMT -->
|
||
</html> |