# Completing the Square

## Theorem

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

Then:

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

This process is known as completing the square.

## Proof

 $\ds a x^2 + b x + c$ $=$ $\ds \frac {4 a^2 x^2 + 4 a b x + 4 a c} {4 a}$ multiplying top and bottom by $4 a$ $\ds$ $=$ $\ds \frac {4 a^2 x^2 + 4 a b x + b^2 + 4 a c - b^2} {4 a}$ $\ds$ $=$ $\ds \frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}$

$\blacksquare$

## Also presented as

This result can also be presented in the form:

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

## Historical Note

The technique of Completing the Square was known to the ancient Babylonians as early as $1600$ BCE.