Completing the Square

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Completing the square writes a quadratic x² + bx + c in the form (x + p)² + q. Halve the coefficient of x to get p, then subtract p² and add the original constant: x² + 8x + 3 = (x + 4)² − 16 + 3 = (x + 4)² − 13. This form solves quadratics exactly and shows the turning point, which for (x + p)² + q is (−p, q).

Completing the square

x² + bx + c = (x + b/2)² − (b/2)² + c
(x + p)² + q has minimum value q, when x = −p
Turning point of y = (x + p)² + q: (−p, q)
To solve: rearrange to (x + p)² = k, then square-root both sides (±)

How to complete the square

Halve the coefficient of x; that number goes inside the bracket. Expanding the bracket gives the right x² and x terms but also an extra constant, p², so subtract it and add the original constant. For x² + 8x + 3: half of 8 is 4, (x + 4)² = x² + 8x + 16, so x² + 8x + 3 = (x + 4)² − 16 + 3 = (x + 4)² − 13.

Solving by completing the square

Complete the square, then rearrange to get the bracket on its own: (x + p)² = k. Take the square root of both sides, remembering ±, and solve. For x² − 10x + 16 = 0: (x − 5)² − 9 = 0, so (x − 5)² = 9, x − 5 = ±3, and x = 8 or x = 2.

Exact answers in surd form

Completing the square gives exact answers even when the quadratic does not factorise. For x² + 4x − 3 = 0: (x + 2)² − 7 = 0, so (x + 2)² = 7, x + 2 = ±√7 and x = −2 ± √7. Questions often ask for answers in this surd form.

The turning point

A square is never negative, so (x + p)² + q is smallest when the bracket is zero, at x = −p, and its minimum value is q. The graph of y = (x + p)² + q therefore has its turning point at (−p, q). For y = x² − 6x + 14 = (x − 3)² + 5, the minimum point is (3, 5).

When the coefficient of x² is not 1

Factor the coefficient out of the x² and x terms first, complete the square inside the bracket, then multiply back out. 2x² + 12x + 7 = 2(x² + 6x) + 7 = 2[(x + 3)² − 9] + 7 = 2(x + 3)² − 11.

Worked examples

Example 1. Write x² + 8x + 3 in the form (x + p)² + q.

Half of 8 is 4: (x + 4)² = x² + 8x + 16.
Subtract 16 and add 3.

Answer: (x + 4)² − 13.

Example 2. Solve x² − 10x + 16 = 0 by completing the square.

(x − 5)² − 25 + 16 = 0, so (x − 5)² = 9.
x − 5 = ±3.

Answer: x = 8 or x = 2.

Example 3. Solve x² + 4x − 3 = 0, giving your answers in surd form.

(x + 2)² − 4 − 3 = 0, so (x + 2)² = 7.
x + 2 = ±√7.

Answer: x = −2 ± √7.

Example 4. Find the coordinates of the turning point of y = x² − 6x + 14.

(x − 3)² − 9 + 14 = (x − 3)² + 5.
The minimum is when x = 3, giving y = 5.

Answer: (3, 5).

Example 5. Write 2x² + 12x + 7 in the form a(x + p)² + q.

Factor 2 from the x terms: 2(x² + 6x) + 7.
x² + 6x = (x + 3)² − 9, so 2[(x + 3)² − 9] + 7.

Answer: 2(x + 3)² − 11.

Common mistakes

Using b instead of b/2. x² + 8x needs (x + 4)², not (x + 8)².
Adding p² instead of subtracting. (x + 4)² is 16 too big, so subtract 16.
Forgetting ± when square-rooting. (x − 5)² = 9 gives x − 5 = 3 or −3.
Turning point sign errors. (x − 3)² + 5 has turning point (3, 5), not (−3, 5).
Factoring a out of the constant too. Take a out of the x² and x terms only.

How completing the square is examined

Completing the square is a Higher tier topic on the 1MA1 GCSE and the 4MA1 International GCSE. Typical questions ask you to:

write a quadratic in the form (x + p)² + q
use the completed square to solve an equation, often in surd form
find the turning point or minimum value of a quadratic
show that a quadratic can be written in a given form

Completed-square questions with full answers are part of the quadratics worksheet, and the revision engine for GCSE algebra produces new quadratics on demand.

This builds on quadratic equations. At A-Level, see completing the square with negative a.