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Quadratic Formula Derivation

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The standard form of a quadratic equation is:

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a x^{2} + b x + c = 0

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Here, $a, b, c \in ℝ$ , and $a \neq 0$

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We begin by first dividing both sides by the coefficient $a$

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⟹ x^{2} + \frac{b}{a} x + \frac{c}{a} = 0

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We can rearrange the equation:

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x^{2} + \frac{b}{a} x = - \frac{c}{a}

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We can then use the method of completing the square. (Maths is Fun has a really good explanation for this technique)

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x^{2} + \frac{b}{a} x + (\frac{b}{2 a})^{2} = \frac{- c}{a} + (\frac{b}{2 a})^{2}

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On our LHS, we can clearly recognize that it is the expanded form of $(x + d)^{2}$ i.e $x^{2} + 2 x \cdot d + d^{2}$

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⟹ (x + \frac{b}{2 a})^{2} = \frac{- c}{a} + \frac{b^{2}}{4 a^{2}} = \frac{- 4 a c + b^{2}}{4 a^{2}}

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Taking the square root of both sides

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\begin{matrix} x + \frac{b}{2 a} & = \frac{\sqrt{- 4 a c + b^{2}}}{2 a} \\ x & = \frac{\pm \sqrt{- 4 a c + b^{2}} - b}{2 a} \\ = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{matrix}

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This gives you the world famous quadratic formula:

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x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

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+ +

Quadratic Formula Derivation

The standard form of a quadratic equation is:

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- +

Quadratic Formula Derivation

The standard form of a quadratic equation is:

@@ -76,14 +94,15 @@

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

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-- cgit v1.2.3 From de19543d7fb44d343b052dc9b34ede78620c4a46 Mon Sep 17 00:00:00 2001 From: Navan Chauhan Date: Wed, 27 Mar 2024 23:36:55 -0600 Subject: Generate --- .../2024-03-26-Derivation-of-the-Quadratic-Equation.html | 16 ++++++++++++++-- 1 file changed, 14 insertions(+), 2 deletions(-) (limited to 'docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html') 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 index bb2bc8e..27ad846 100644 --- a/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html +++ b/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html @@ -46,7 +46,7 @@ + +

-- cgit v1.2.3 From a982ceab0b45609991179b3020a00260eed6f798 Mon Sep 17 00:00:00 2001 From: Navan Chauhan Date: Wed, 27 Mar 2024 23:45:59 -0600 Subject: css --- docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html | 1 + 1 file changed, 1 insertion(+) (limited to 'docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html') 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 index 27ad846..33d09ec 100644 --- a/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html +++ b/docs/posts/2024-03-26-Derivation-of-the-Quadratic-Equation.html @@ -5,6 +5,7 @@ + Quadratic Formula Derivation -- cgit v1.2.3