Derivation of the Quadratic Formula

Starting from the quadratic equation, we will complete the square to get $x$ by itself and use the square root method to solve.

$a x^2 + b x + c = 0$

$a x^2 + b x = -c$

Divide through by $a$

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

Complete the square by multiplying $\frac{b}{a}$ by $\frac{1}{2}$ , squaring the result, and adding it to each side of the equation.

$x^2 + \frac{b}{a} x + \frac{b^2}{4a^2}= -\frac{c}{a} + \frac{b^2}{4a^2}$

Factoring.

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

Simplifying right hand side.

$\left( x + \frac{b}{2a} \right) ^ 2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2}$

$\left( x + \frac{b}{2a} \right) ^ 2 = \frac{b^2}{4a^2} - \frac{4ac}{4a^2}$

$\left( x + \frac{b}{2a} \right) ^ 2 = \frac{b^2-4ac}{4a^2}$

Take the square root of each side.

$\sqrt{ \left( x + \frac{b}{2a} \right) ^ 2 } = \pm \sqrt{ \frac{b^2-4ac}{4a^2}}$

Simplifying.

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

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

Getting $x$ by itself and simplifying.

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

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

We see that the quadratic formula is found by applying the methods of completing the square and solving by square root to the quadratic equation.