A level-readyAll subjects · Mathematics · WMA11 (P1)
Completing the square
Completing the square rewrites a quadratic as a(x + p)2 + q. It shows the turning point at once and gives another way to solve.
When a = 1
x2 + bx + c = (x + b/2)2 - (b/2)2 + c.
Halve the x coefficient, square the bracket, then subtract the square you added.
x2 + 8x + 3 = (x + 4)2 - 16 + 3 = (x + 4)2 - 13.
When a is not 1
Take a out of the x2 and x terms first:
2x2 + 12x + 5 = 2(x2 + 6x) + 5 = 2((x + 3)2 - 9) + 5 = 2(x + 3)2 - 13.
What it tells you
For y = a(x + p)2 + q:
- the turning point is (-p, q)
- if a > 0, q is the minimum value; if a < 0, q is the maximum value
- to solve a(x + p)2 + q = 0, rearrange and square root, remembering +/-
Worked example
Solve x2 - 6x + 2 = 0 by completing the square.
- x2 - 6x + 2 = (x - 3)2 - 9 + 2 = (x - 3)2 - 7
- (x - 3)2 = 7
- x - 3 = +/- sqrt(7)
Answer: x = 3 +/- sqrt(7)
Check yourself
- x2 + 8x + 3 = (x + p)2 + q. Find q.
Show answer
-13. (x + 4)2 = x2 + 8x + 16, so subtract 16: q = 3 - 16 = -13.
- What is the turning point of y = (x - 5)2 + 2?
- (-5, 2)
- (5, -2)
- (5, 2)
- (2, 5)
Show answer
(5, 2). For (x + p)2 + q the turning point is (-p, q). Here p = -5 and q = 2.
- Find the minimum value of x2 - 10x + 30.
Show answer
5. x2 - 10x + 30 = (x - 5)2 + 5. The square is never negative, so the minimum is 5.
- 3x2 - 12x + 7 = a(x + p)2 + q. Find q.
Show answer
-5. 3(x2 - 4x) + 7 = 3((x - 2)2 - 4) + 7 = 3(x - 2)2 - 5, so q = -5.
- What is the maximum value of y = 4 - (x + 1)2, and where does it occur?
- 4, when x = 1
- 4, when x = -1
- -1, when x = 4
- -4, when x = -1
Show answer
4, when x = -1. (x + 1)2 is never negative, so y is largest when the bracket is 0, at x = -1. Then y = 4.
- Solve x2 - 6x + 2 = 0, giving exact answers.
- x = -3 +/- sqrt(7)
- x = 3 +/- sqrt(11)
- x = 3 +/- sqrt(2)
- x = 3 +/- sqrt(7)
Show answer
x = 3 +/- sqrt(7). (x - 3)2 - 7 = 0, so x = 3 +/- sqrt(7).
- x2 + 6x + k = (x + 3)2 + 4. Find k.
Show answer
13. (x + 3)2 + 4 = x2 + 6x + 9 + 4, so k = 13.
Sources
- Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
- Pearson Edexcel IAL Pure Mathematics 1 scheme of work
Last updated 7 November 2026. Spotted a mistake? Tell us.