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---
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:
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
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