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| author | Navan Chauhan <navanchauhan@gmail.com> | 2024-04-03 09:36:58 -0600 |
|---|---|---|
| committer | GitHub <noreply@github.com> | 2024-04-03 09:36:58 -0600 |
| commit | 84ad5d8360fb0683f12f9bae8f29d6b36964dde3 (patch) | |
| tree | 64bb1d9675ac3d6a3604d512311a754b08c53be2 /docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html | |
| parent | 0e4b5fcc164cd8432678bdffd58b8c123acee186 (diff) | |
| parent | d00b2da99ef2d498074a6e4eb8d80995cdfc4476 (diff) | |
Merge branch 'vanilla-ssg' into dependabot/pip/jinja2-3.1.3
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| -rw-r--r-- | docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html | 122 |
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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..33d09ec --- /dev/null +++ b/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html @@ -0,0 +1,122 @@ +<!DOCTYPE html> +<html lang="en"> +<head> + + <meta http-equiv="X-UA-Compatible" content="IE=edge"> + <meta http-equiv="content-type" content="text/html; charset=utf-8"> + <meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1"> + <meta name="theme-color" content="#6a9fb5"> + + <title>Quadratic Formula Derivation</title> + + <!-- + <link rel="stylesheet" href="https://unpkg.com/latex.css/style.min.css" /> + --> + + <link rel="stylesheet" href="/assets/c-hyde.css" /> + + <link rel="stylesheet" href="http://fonts.googleapis.com/css?family=PT+Sans:400,400italic,700|Abril+Fatface"> + + <link rel="stylesheet" href="/assets/main.css" /> + <meta charset="utf-8"> + <meta name="viewport" content="width=device-width, initial-scale=1.0"> + <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 class="theme-base-0d"> + <div class="sidebar"> + <div class="container sidebar-sticky"> + <div class="sidebar-about"> + <h1><a href="/">Navan</a></h1> + <p class="lead" id="random-lead">Alea iacta est.</p> + </div> + + <ul class="sidebar-nav"> + <li><a class="sidebar-nav-item" href="/about/">about/links</a></li> + <li><a class="sidebar-nav-item" href="/posts/">posts</a></li> + <li><a class="sidebar-nav-item" href="/3D-Designs/">3D designs</a></li> + <li><a class="sidebar-nav-item" href="/feed.rss">RSS Feed</a></li> + <li><a class="sidebar-nav-item" href="/colophon/">colophon</a></li> + </ul> + <div class="copyright"><p>© 2019-2024. Navan Chauhan <br> <a href="/feed.rss">RSS</a></p></div> + </div> +</div> + +<script> +let phrases = [ + "Something Funny", "Veni, vidi, vici", "Alea iacta est", "In vino veritas", "Acta, non verba", "Castigat ridendo mores", + "Cui bono?", "Memento vivere", "अहम् ब्रह्मास्मि", "अनुगच्छतु प्रवाहं", "चरन्मार्गान्विजानाति", "coq de cheval", "我愛啤酒" + ]; + +let new_phrase = phrases[Math.floor(Math.random()*phrases.length)]; + +let lead = document.getElementById("random-lead"); +lead.innerText = new_phrase; +</script> + <div class="content container"> + + <div class="post"> + <h1 id="quadratic-formula-derivation">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> + + </div> + <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> + + </div> + <script src="assets/manup.min.js"></script> + <script src="/pwabuilder-sw-register.js"></script> +</body> +</html>
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