Completing the square: method, examples and solutions
Rewrite quadratics in completed-square form to solve equations and find the turning point, with step-by-step examples for GCSE Higher and Algebra 2.
GCSEA-LevelHigh SchoolCurriculum fit: GCSE Maths Algebra (Higher tier). US: Algebra I and II, completing the square (Seeing Structure in Expressions; Reasoning with Equations).
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What is it?
Completing the square rewrites a quadratic such as x² + 6x + 2 in the form (x + p)² + q. This form is useful because it shows the turning point of the graph and lets you solve equations that do not factorise.
The method relies on the fact that (x + p)² expands to x² + 2px + p².
Key formulas and ideas
- x² + bx + c = (x + b/2)² + c − (b/2)²
- ax² + bx + c = a(x + b/2a)² + c − b²/4a
- The turning point of y = (x + p)² + q is (−p, q).
Worked examples
Example 1: Rewrite
Question: Write x² + 6x + 2 in the form (x + p)² + q.
- Halve the coefficient of x: 6 ÷ 2 = 3, so p = 3.
- (x + 3)² = x² + 6x + 9, which has 9 too many for a constant of 2.
- Subtract 7: x² + 6x + 2 = (x + 3)² − 7.
Answer: (x + 3)² − 7
Example 2: Solve
Question: Solve x² + 6x + 2 = 0, leaving surds.
- Complete the square: (x + 3)² − 7 = 0.
- (x + 3)² = 7.
- x + 3 = ±√7, so x = −3 ± √7.
Answer: x = −3 + √7 or x = −3 − √7
Example 3: Leading coefficient not 1
Question: Write 2x² + 8x + 3 in the form a(x + p)² + q.
- Factor 2 from the x terms: 2(x² + 4x) + 3.
- Complete the square inside: 2((x + 2)² − 4) + 3.
- Expand: 2(x + 2)² − 8 + 3 = 2(x + 2)² − 5.
Answer: 2(x + 2)² − 5
Common mistakes
- Using b instead of b/2 inside the bracket.
- Adding the correction instead of subtracting it.
- Forgetting the ± when taking a square root.
- Not taking out the leading coefficient first.
Practice questions
Frequently asked questions
Why bother when there is a quadratic formula?
Completing the square also gives the turning point, and the quadratic formula itself is derived from it.
What is the turning point of y = (x − 4)² + 1?
(4, 1), because the bracket (x − 4) is zero when x = 4.
Related guides
Solving quadratic equations: factorising and the formula
Solve quadratics by factorising and with the quadratic formula, using worked examples. Covers GCSE quadratic equations and Algebra 1 and 2 methods.
GCSEA-LevelHigh SchoolQuadratic graphs: roots, turning points and symmetry
Sketch parabolas and find the roots, y-intercept, line of symmetry and turning point (vertex) of a quadratic, with step-by-step worked examples.
GCSEA-LevelHigh SchoolSurds (radicals): simplifying and rationalising
Simplify surds, calculate with them and rationalise denominators, with step-by-step examples. UK surds and US radicals, for GCSE Higher and Algebra 2.
GCSEA-LevelHigh SchoolSources and review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.