Completing the square

#### Tier: #Higher

Description

Completing the square rewrites $ax^2 + bx + c$ in the form $a(x + p)^2 + q$. It is used to solve quadratics, find turning points, and derive the quadratic formula.

Method (for $a = 1$): $$x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c$$

Example: Complete the square for $x^2 + 6x + 7$. $$\left(x + 3\right)^2 - 9 + 7 = (x+3)^2 - 2$$

Solving by completing the square: $$(x+3)^2 - 2 = 0 \implies (x+3)^2 = 2 \implies x = -3 \pm \sqrt{2}$$

Turning point: $(x+3)^2 - 2$ has minimum at $(-3, -2)$.

For $a \neq 1$: factorise out $a$ first, then complete the square in the bracket.

Common error: forgetting to subtract $\left(\frac{b}{2}\right)^2$ after adding it inside the bracket — the expression must stay equivalent to the original.

Links

Manipulating algebraic fractions Factorising quadratics Solving quadratics graphically