# Quadratic Formula Derivation

The standard form of a quadratic equation is:

$$a{x}^{2}+bx+c=0$$Here, $a,b,c\in \mathbb{R}$, and $a\ne 0$

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

$$\u27f9{x}^{2}+\frac{b}{a}x+\frac{c}{a}=0$$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 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 ${x}^{2}+2x\xb7d+{d}^{2}$

$$\u27f9(x+\frac{b}{2a}{)}^{2}=\frac{-c}{a}+\frac{{b}^{2}}{4{a}^{2}}=\frac{-4ac+{b}^{2}}{4{a}^{2}}$$Taking the square root of both sides

$$\begin{array}{cc}\hfill x+\frac{b}{2a}& =\frac{\sqrt{-4ac+{b}^{2}}}{2a}\hfill \\ \hfill x& =\frac{\pm \sqrt{-4ac+{b}^{2}}-b}{2a}\hfill \\ \hfill & =\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}\hfill \end{array}$$This gives you the world famous quadratic formula:

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