# Completing the Square

## Theorem

Let $a,b,c,x$ be real numbers with $a \neq 0$.

Then

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

This process is known as completing the square.

## Proof

 $\displaystyle a x^2 + b x + c$ $=$ $\displaystyle \frac {4 a^2 x^2 + 4 a b x + 4 a c} {4 a}$ $\quad$ multiplying top and bottom by $4 a$ $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {4 a^2 x^2 + 4 a b x + b^2 + 4 a c - b^2} {4 a}$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {\left({2 a x + b}\right)^2 + 4 a c - b^2} {4 a}$ $\quad$ $\quad$

$\blacksquare$