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| author | Navan Chauhan <navanchauhan@gmail.com> | 2024-03-26 18:21:29 -0600 |
|---|---|---|
| committer | Navan Chauhan <navanchauhan@gmail.com> | 2024-03-26 18:21:29 -0600 |
| commit | aae00025bd8bff04de90b22b2472aed8a232f476 (patch) | |
| tree | 42dcca0448ac2e87e028b4890942977e31dc5d9f | |
| parent | 37661080a111768e565ae53299c4796ebe711a71 (diff) | |
post testing latex extra
| -rw-r--r-- | Content/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.md | 55 | ||||
| -rw-r--r-- | Resources/images/opengraph/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.png | bin | 0 -> 21809 bytes | |||
| -rw-r--r-- | docs/feed.rss | 60 | ||||
| -rw-r--r-- | docs/images/opengraph/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.png | bin | 0 -> 21809 bytes | |||
| -rw-r--r-- | docs/index.html | 24 | ||||
| -rw-r--r-- | docs/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html | 10 | ||||
| -rw-r--r-- | docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html | 90 | ||||
| -rw-r--r-- | docs/posts/index.html | 11 | ||||
| -rw-r--r-- | docs/tags/mathematics.html | 11 |
9 files changed, 232 insertions, 29 deletions
diff --git a/Content/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.md b/Content/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.md new file mode 100644 index 0000000..0435a6c --- /dev/null +++ b/Content/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.md @@ -0,0 +1,55 @@ +--- +date: 2024-03-26 15:36 +description: Quick derivation of the quadratic equation by completing the square +tags: mathematics +--- + +# Quadratic Formula Derivation + +The standard form of a quadratic equation is: + +$$ +ax^2 + bx + c = 0 +$$ + +Here, $a, b, c \in \mathbb{R}$, and $a \neq 0$ + +We begin by first dividing both sides by the coefficient $a$ + +$$ +\implies x^2 + \frac{b}{a}x + \frac{c}{a} = 0 +$$ + +We can rearrange the equation: + +$$ +x^2 + \frac{b}{a}x = - \frac{c}{a} +$$ + +We can then use the method of completing the square. ([Maths is Fun](https://www.mathsisfun.com/algebra/completing-square.html) has a really good explanation for this technique) + +$$ +x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 = \frac{-c}{a} + (\frac{b}{2a})^2 +$$ + +On our LHS, we can clearly recognize that it is the expanded form of $(x + d)^2$ i.e $x^2 + 2x\cdot d + d^2$ + +$$ +\implies (x + \frac{b}{2a})^2 = \frac{-c}{a} + \frac{b^2}{4a^2} = \frac{-4ac + b^2}{4a^2} +$$ + +Taking the square root of both sides + +$$ +\begin{align*} +x + \frac{b}{2a} &= \frac{\sqrt{-4ac + b^2}}{2a} \\ +x &= \frac{\pm \sqrt{-4ac + b^2} - b}{2a} \\ +&= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} +\end{align*} +$$ + +This gives you the world famous quadratic formula: + +$$ +x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} +$$ diff --git a/Resources/images/opengraph/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.png b/Resources/images/opengraph/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.png Binary files differnew file mode 100644 index 0000000..2464364 --- /dev/null +++ b/Resources/images/opengraph/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.png diff --git a/docs/feed.rss b/docs/feed.rss index 12e9f8d..12b90df 100644 --- a/docs/feed.rss +++ b/docs/feed.rss @@ -4,8 +4,8 @@ <title>Navan's Archive</title> <description>Rare Tips, Tricks and Posts</description> <link>https://web.navan.dev/</link><language>en</language> - <lastBuildDate>Thu, 21 Mar 2024 14:27:28 -0000</lastBuildDate> - <pubDate>Thu, 21 Mar 2024 14:27:28 -0000</pubDate> + <lastBuildDate>Tue, 26 Mar 2024 18:20:37 -0000</lastBuildDate> + <pubDate>Tue, 26 Mar 2024 18:20:37 -0000</pubDate> <ttl>250</ttl> <atom:link href="https://web.navan.dev/feed.rss" rel="self" type="application/rss+xml"/> @@ -557,9 +557,7 @@ creating<span class="w"> </span>a<span class="w"> </span>DOS<span class="w"> </s <script src="https://cdn.jsdelivr.net/npm/mathjax@4.0.0-beta.4/input/tex/extensions/noerrors.js" charset="UTF-8"></script> -<p>$$ -y = ax^3 + bx^2 + cx + d -$$</p> +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></mrow></math> <h3>Optimizer Selection & Training</h3> @@ -584,9 +582,7 @@ $$</p> <p>Our loss function is Mean Squared Error (MSE):</p> -<p>$$ -= \frac{1}{n} \sum_{i=1}^{n}{(Y_i - \hat{Y_i})^2} -$$</p> +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mi>Y</mi><mi>_</mi><mi>i</mi><mo>−</mo><mover><mrow><mi>Y</mi><mi>_</mi><mi>i</mi></mrow><mo stretchy="false">^</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mrow></math> <p>Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false" style="math-style:normal;math-depth:0;">^</mo></mover></math> is the predicted value and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub></math> is the actual value</p> @@ -726,7 +722,7 @@ $$</p> <p>How would you modify this code to use another type of nonlinear regression? Say, </p> -<p>$$ y = ab^x $$</p> +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>b</mi><mi>x</mi></msup></mrow></math> <p>Hint: Your loss calculation would be similar to:</p> @@ -3188,6 +3184,52 @@ values using the X values. We then plot it to compare the actual data and predic <item> <guid isPermaLink="true"> + https://web.navan.dev/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html + </guid> + <title> + Quadratic Formula Derivation + </title> + <description> + Quick derivation of the quadratic equation by completing the square + </description> + <link>https://web.navan.dev/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html</link> + <pubDate>Tue, 26 Mar 2024 15:36:00 -0000</pubDate> + <content:encoded><![CDATA[<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> +]]></content:encoded> + </item> + + <item> + <guid isPermaLink="true"> https://web.navan.dev/posts/2022-08-05-Why-You-No-Host.html </guid> <title> diff --git a/docs/images/opengraph/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.png b/docs/images/opengraph/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.png Binary files differnew file mode 100644 index 0000000..2464364 --- /dev/null +++ b/docs/images/opengraph/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.png diff --git a/docs/index.html b/docs/index.html index 0a3070a..c462f4b 100644 --- a/docs/index.html +++ b/docs/index.html @@ -50,6 +50,17 @@ <h2>Recent Posts</h2> <ul> + <li><a href="/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html">Quadratic Formula Derivation</a></li> + <ul> + <li>Quick derivation of the quadratic equation by completing the square</li> + <li>Published On: 2024-03-26 15:36</li> + <li>Tags: + + <a href='/tags/mathematics.html'>mathematics</a> + + </ul> + + <li><a href="/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html">Polynomial Regression Using TensorFlow 2.x</a></li> <ul> <li>Predicting n-th degree polynomials using TensorFlow 2.x</li> @@ -106,19 +117,6 @@ </ul> - <li><a href="/posts/2024-01-05-hello-20224.html">Hello 2024</a></li> - <ul> - <li>Recap of 2023, and my goals for 2024</li> - <li>Published On: 2024-01-05 23:16</li> - <li>Tags: - - <a href='/tags/General.html'>General</a>, - - <a href='/tags/Ramblings.html'>Ramblings</a> - - </ul> - - </ul> <b>For all posts go to <a href="/posts">Posts</a></b> diff --git a/docs/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html b/docs/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html index 7a25daf..ab46ec7 100644 --- a/docs/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html +++ b/docs/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html @@ -107,9 +107,7 @@ <script src="https://cdn.jsdelivr.net/npm/mathjax@4.0.0-beta.4/input/tex/extensions/noerrors.js" charset="UTF-8"></script> -<p>$$ -y = ax^3 + bx^2 + cx + d -$$</p> +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></mrow></math> <h3>Optimizer Selection & Training</h3> @@ -134,9 +132,7 @@ $$</p> <p>Our loss function is Mean Squared Error (MSE):</p> -<p>$$ -= \frac{1}{n} \sum_{i=1}^{n}{(Y_i - \hat{Y_i})^2} -$$</p> +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mrow><mo stretchy="false">(</mo><mi>Y</mi><mi>_</mi><mi>i</mi><mo>−</mo><mover><mrow><mi>Y</mi><mi>_</mi><mi>i</mi></mrow><mo stretchy="false">^</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mrow></math> <p>Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><msub><mi>Y</mi><mi>i</mi></msub><mo stretchy="false" style="math-style:normal;math-depth:0;">^</mo></mover></math> is the predicted value and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mi>i</mi></msub></math> is the actual value</p> @@ -276,7 +272,7 @@ $$</p> <p>How would you modify this code to use another type of nonlinear regression? Say, </p> -<p>$$ y = ab^x $$</p> +<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>b</mi><mi>x</mi></msup></mrow></math> <p>Hint: Your loss calculation would be similar to:</p> 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>
\ No newline at end of file diff --git a/docs/posts/index.html b/docs/posts/index.html index d886b19..40b6a92 100644 --- a/docs/posts/index.html +++ b/docs/posts/index.html @@ -52,6 +52,17 @@ <ul> + <li><a href="/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html">Quadratic Formula Derivation</a></li> + <ul> + <li>Quick derivation of the quadratic equation by completing the square</li> + <li>Published On: 2024-03-26 15:36</li> + <li>Tags: + + <a href='/tags/mathematics.html'>mathematics</a> + + </ul> + + <li><a href="/posts/2024-03-21-Polynomial-Regression-in-TensorFlow-2.html">Polynomial Regression Using TensorFlow 2.x</a></li> <ul> <li>Predicting n-th degree polynomials using TensorFlow 2.x</li> diff --git a/docs/tags/mathematics.html b/docs/tags/mathematics.html index b948423..eb5201c 100644 --- a/docs/tags/mathematics.html +++ b/docs/tags/mathematics.html @@ -49,6 +49,17 @@ <ul> + <li><a href="/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html">Quadratic Formula Derivation</a></li> + <ul> + <li>Quick derivation of the quadratic equation by completing the square</li> + <li>Published On: 2024-03-26 15:36</li> + <li>Tags: + + <a href='/tags/mathematics.html'>mathematics</a> + + </ul> + + <li><a href="/posts/2023-04-30-n-body-simulation.html">n-body solution generator</a></li> <ul> <li>n-body solution generator and solver</li> |
