For a monic quadratic, halve the coefficient of x, then square that half. Adding and subtracting this amount preserves the expression: x2+px+q=(x+ p2)2+q- p24. Example Solve x2-6x+2=0. x2-6x+9-9+2=0 (x-3)2=7 x-3=+- 7 x=3+- 7. 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.

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

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

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 \(x=3\).

Quadratic Functions, Equations and Inequalities

Cambridge IGCSE Mathematics 0580. Initial catalogue covers the assessed demands of February/March 2026 Paper 22; further Core and Extended mappings require paper-level scope review.

Completing the square

Completing the square