# Completing the Square

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## Contents

## 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

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

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {4 a^2 x^2 + 4 a b x + b^2 + 4 a c - b^2} {4 a}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {\paren {2 a x + b}^2 + 4 a c - b^2} {4 a}\) |

$\blacksquare$

## Also see

## Historical Note

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

## Sources

- 2004: Ian Stewart:
*Galois Theory*(3rd ed.) ... (previous) ... (next): Historical Introduction: Polynomial Equations