Completing the square

For a monic quadratic, halve the coefficient of \(x\), then square that half. Adding and subtracting this amount preserves the expression: \(x^2+px+q=\left(x+\dfrac p2\right)^2+q-\dfrac{p^2}{4}\).

Example

Solve \(x^2-6x+2=0\).

\(\begin{aligned}x^2-6x+9-9+2&=0\\(x-3)^2&=7\\x-3&=\pm\sqrt7\\x&=3\pm\sqrt7.\end{aligned}\)

The two signs are needed because both a positive and a negative number can have square \(7\). The same completed form, \(y=(x-3)^2-7\), shows that its graph has minimum \(-7\) at \(x=3\).