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MathematicsGCSE Higher & A Level

What is completing the square?

Completing the square rewrites a quadratic in the form a(x + p)² + q, revealing the turning point and allowing exact solutions.

Completing the square explained

Halve the coefficient of xx, square it, and balance the expression. If a≠1a \neq 1, factor it out of the first two terms before you start.
The completed form gives the vertex at (−p,q)(-p, q), the line of symmetry x=−px = -p, and the minimum or maximum value directly.

Key formula

x2+bx+c=(x+b2)2−b24+cx^{2} + bx + c = \left(x + \tfrac{b}{2}\right)^{2} - \tfrac{b^{2}}{4} + c

Worked example

x2+6x+1=(x+3)2−8x^{2} + 6x + 1 = (x + 3)^{2} - 8, so the minimum point is (−3,−8)(-3, -8).

Examiner tip

If the question says 'hence' after completing the square, use your completed form — solving from scratch usually scores zero for that part.

Completing the square: common questions

Why is completing the square useful?
It gives the turning point without calculus, proves maximum or minimum values, and derives the quadratic formula itself.
How do I complete the square when a is not 1?
Factor a out of the x² and x terms first, complete the square inside the bracket, then expand the a back through.

Related terms

Quadratic formula

The quadratic formula solves any equation of the form ax² + bx + c = 0, including those that will not factorise.

Discriminant

The discriminant, b² − 4ac, tells you how many real roots a quadratic equation has without solving it.

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