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diff --git a/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html b/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html new file mode 100644 index 0000000..6f02f7c --- /dev/null +++ b/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html @@ -0,0 +1,90 @@ +<!DOCTYPE html> +<html lang="en"> +<head> + + <link rel="stylesheet" href="https://unpkg.com/latex.css/style.min.css" /> + <link rel="stylesheet" href="/assets/main.css" /> + <meta charset="utf-8"> + <meta name="viewport" content="width=device-width, initial-scale=1.0"> + <title>Quadratic Formula Derivation</title> + <meta name="og:site_name" content="Navan Chauhan" /> + <link rel="canonical" href="https://web.navan.dev/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html" /> + <meta name="twitter:url" content="https://web.navan.dev/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html /> + <meta name="og:url" content="https://web.navan.dev/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html" /> + <meta name="twitter:title" content="Quadratic Formula Derivation" /> + <meta name="og:title" content="Quadratic Formula Derivation" /> + <meta name="description" content="Quick derivation of the quadratic equation by completing the square" /> + <meta name="twitter:description" content="Quick derivation of the quadratic equation by completing the square" /> + <meta name="og:description" content="Quick derivation of the quadratic equation by completing the square" /> + <meta name="twitter:card" content="summary_large_image" /> + <meta name="viewport" content="width=device-width, initial-scale=1.0" /> + <link rel="shortcut icon" href="/images/favicon.png" type="image/png" /> + <link rel="alternate" href="/feed.rss" type="application/rss+xml" title="Subscribe to Navan Chauhan" /> + <meta name="twitter:image" content="https://web.navan.dev/images/opengraph/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.png" /> + <meta name="og:image" content="https://web.navan.dev/images/opengraph/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.png" /> + <meta name="google-site-verification" content="LVeSZxz-QskhbEjHxOi7-BM5dDxTg53x2TwrjFxfL0k" /> + <script data-goatcounter="https://navanchauhan.goatcounter.com/count" + async src="//gc.zgo.at/count.js"></script> + <script defer data-domain="web.navan.dev" src="https://plausible.io/js/plausible.js"></script> + <link rel="manifest" href="/manifest.json" /> + +</head> +<body> + <center><nav style="display: block;"> +| +<a href="/">home</a> | +<a href="/about/">about/links</a> | +<a href="/posts/">posts</a> | +<a href="/3D-Designs/">3D designs</a> | +<!--<a href="/publications/">publications</a> |--> +<!--<a href="/repo/">iOS repo</a> |--> +<a href="/feed.rss">RSS Feed</a> | +</nav> +</center> + +<main> + + <h1>Quadratic Formula Derivation</h1> + +<p>The standard form of a quadratic equation is:</p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math> + +<p>Here, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>∈</mo><mi>ℝ</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow></math></p> + +<p>We begin by first dividing both sides by the coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi></mrow></math></p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>⟹</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math> + +<p>We can rearrange the equation:</p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac></mrow></math> + +<p>We can then use the method of completing the square. (<a rel="noopener" target="_blank" href="https://www.mathsisfun.com/algebra/completing-square.html">Maths is Fun</a> has a really good explanation for this technique)</p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mo>−</mo><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>+</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></math> + +<p>On our LHS, we can clearly recognize that it is the expanded form of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>d</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></math> i.e <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>·</mi><mi>d</mi><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></mrow></math></p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>⟹</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mo>−</mo><mi>c</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac></mrow></math> + +<p>Taking the square root of both sides</p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable displaystyle="true" rowspacing="3pt" columnspacing="0em 2em"><mtr><mtd columnalign="right"><mi>x</mi><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd><mtd columnalign="left"><mi /><mo>=</mo><mfrac><mrow><msqrt><mrow><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mi /><mo>=</mo><mfrac><mrow><mi>±</mi><msqrt><mrow><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></msqrt><mo>−</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><mi /><mo>=</mo><mfrac><mrow><mo>−</mo><mi>b</mi><mi>±</mi><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mtd></mtr></mtable></mrow></math> + +<p>This gives you the world famous quadratic formula:</p> + +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><mo>−</mo><mi>b</mi><mi>±</mi><msqrt><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>a</mi><mi>c</mi></mrow></msqrt></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mrow></math> + + <blockquote>If you have scrolled this far, consider subscribing to my mailing list <a href="https://listmonk.navan.dev/subscription/form">here.</a> You can subscribe to either a specific type of post you are interested in, or subscribe to everything with the "Everything" list.</blockquote> + <script data-isso="https://comments.navan.dev/" + src="https://comments.navan.dev/js/embed.min.js"></script> + <section id="isso-thread"> + <noscript>Javascript needs to be activated to view comments.</noscript> + </section> +</main> + + <script src="assets/manup.min.js"></script> + <script src="/pwabuilder-sw-register.js"></script> +</body> +</html>
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