Completing the square

Algebra

(1MA1) · Higher

ORIGINAL

Solve a quadratic exactly by completing the square — the method examiners want when 'give exact answers' appears.

Verified against Edexcel 1MA1 (2026 spec)

Question

Solve $x^2 + 6x + 2 = 0$ by completing the square. Give your answers in exact (surd) form.

Every step worked, with the reasoning.

1
$x^2 + 6x + 2 = 0$
Start from the equation.
2
$(x + 3)^2 - 9 + 2 = 0$
Halve the coefficient of $x$ ( $6 \div 2 = 3$ ) to get $(x+3)^2$ , then subtract $3^2 = 9$ to keep it equal.
3
$(x + 3)^2 - 7 = 0$
Combine the constants: $-9 + 2 = -7$ .
4
$(x + 3)^2 = 7$
Add 7 to both sides to isolate the square.
5
$x + 3 = \pm\sqrt{7}$
Square-root both sides — keep the $\pm$ or you lose a solution.
6
$x = -3 \pm\sqrt{7}$
Subtract 3 from both sides.

Answer: $x = -3 + \sqrt{7}$ or $x = -3 - \sqrt{7}$

This is how to revise a method, not just read it

Fade the steps out until you can do it cold. Want a set built around exactly what you keep slipping on? Your first lesson is free.