Rewrite a quadratic as (x + p)² + q to read off its turning point, sketch it, or solve it exactly with surds. It is Higher only, aimed at grade 8. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Completing the square rewrites x2+bx+c as (x+2b)2 plus a leftover constant. That form hands you the turning point (−p,q) of y=(x+p)2+q and lets you solve by square-rooting. If a=1, factor a out of the x2 and x terms first, complete inside, then multiply back.
Check by expanding. Exact roots: (x+3)2=11 so x=−3±11. The quadratic formula is this process done in general. Use completed-square form when the turning point or a minimum value is asked; use the formula when you only need 2 d.p. roots in a hurry.
Halve the x-coefficient, square it, add and subtract. x2−10x+30=(x−5)2+5. The turning point of y=(x−h)2+k is (h,k) — flip the sign of h. Solving: (x−h)2=N then x=h±N.
Objectives
Complete the square when a is 1 and when a is not 1.
Read a turning point from the completed form.
Solve and find a max or min from that form.
Key points
Complete the square for x2+bx+c by halving b and subtracting the square of that half.
Handle a coefficient of x2 that is not 1 by factorising it out first.
Read the turning point (−p,q) straight from (x+p)2+q.
Solve a quadratic exactly by isolating the squared bracket and square-rooting both sides.
Worked examples
Example 1
Write x^2+6x+2 in completed-square form.
2 marks
Half of 6 is 3
(x+3)^2 = x^2+6x+9
Subtract 9 and put the 2 back: (x+3)^2-7
Answer: (x+3)^2-7
Common slips
Slip: Half of 6 is 6 The linear coefficient is halved, not copied.
Slip: (x+3)^2 = x^2+6x+3 Squaring the bracket gives 9, not 3.
Slip: Leave it as (x+3)^2+2 You added 9 when you squared, so you must subtract 9.
Example 2
Write 2x^2+8x+5 in the form a(x+p)^2+q.
2 marks
Factor 2 from the x terms: 2(x^2+4x)+5
Half of 4 is 2, so 2[(x+2)^2-4]+5
2(x+2)^2-8+5 = 2(x+2)^2-3
Answer: 2(x+2)^2-3
Common slips
Slip: Factor 2 from everything including the 5 5 is not divisible by 2 in this expression.
Slip: Half of 4 is 4 You halve the coefficient of x inside the bracket.
Slip: 2(x+2)^2-4+5 The -4 is inside the factor of 2, so it becomes -8.
Example 3
Solve 2x^2+4x-2=0 by completing the square.
3 marks
2(x^2+2x)-2=0
2[(x+1)^2-1]-2=0 so 2(x+1)^2-4=0
(x+1)^2=2 so x=-1+√2 or x=-1-√2
Answer: -1-√2 or -1+√2
Common slips
Slip: 2(x^2+2x-2)=0 and stop Keep going until the square is isolated.
Slip: 2(x+1)^2-1=0 The -1 is multiplied by 2, then the original -2 is still there.
Slip: x=2 only Square-rooting gives two answers.
Questions and answers
GCSE Maths completing the square questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Write x^2+10x+3 as a completed square.
Show answer
Answer: (x+5)^2-22
Half of 10 is 5.
(x+5)^2-25+3.
M1 — Half of 10 is 5.
A1 — Answer (x+5)^2-22
Question 2 Easy2 marks
State the turning point of y=2x^2-8x+3.
Show answer
Answer: (2, -5)
Factor 2 from the x terms: 2(x^2-4x)+3.
2[(x-2)^2-4]+3 = 2(x-2)^2-5, and the bracket is zero at x=2.
M1 — Factor 2 from the x terms: 2(x^2-4x)+3.
A1 — Answer (2, -5)
Question 3 Medium2 marks
Write 3x^2-12x+7 as a completed square.
Show answer
Answer: 3(x-2)^2-5
3(x^2-4x)+7.
3[(x-2)^2-4]+7 = 3(x-2)^2-5.
M1 — 3(x^2-4x)+7.
A1 — Answer 3(x-2)^2-5
Question 4 Medium2 marks
Write 2x^2+4x+9 in completed-square form.
Show answer
Answer: 2(x+1)^2+7
2(x^2+2x)+9.
2[(x+1)^2-1]+9.
M1 — 2(x^2+2x)+9.
A1 — Answer 2(x+1)^2+7
Question 5 Hard2 marks
Solve x^2+2x-8=0 by completing the square and state both roots.
Show answer
Answer: -4 or 2
(x+1)^2-1-8=0.
(x+1)^2=9.
M1 — (x+1)^2-1-8=0.
A1 — Answer -4 or 2
Question 6 Hard2 marks
Complete the square for 5-4x-x^2 and state the maximum value.
Show answer
Answer: 9
-(x^2+4x)+5.
9-(x+2)^2. The greatest value is 9.
M1 — -(x^2+4x)+5.
A1 — Answer 9
Question 7 Exam-style5 marks
Solve 2x^2-12x+10=0 by completing the square.
Show answer
Answer: 1 or 5
2(x^2-6x)+10=0.
2[(x-3)^2-9]+10=0.
2(x-3)^2-8=0.
(x-3)^2=4.
x=5 or x=1.
M1 — Factor 2
M1 — Square inside
M1 — Isolate the square
A1 — First root
A1 — Second root
Common mistakes
Most lost marks on this topic are one slip.
Writing (x+b)^2 instead of (x+b/2)^2
Halve the linear coefficient before you square
Forgetting to multiply the adjustment by a
If you factored a, the number you subtracted inside is multiplied by a
Giving one root after a square root
A positive square has two square roots
Spot the slip
Sam writes x^2+6x+2=(x+6)^2+2. What went wrong?
6 was not halved
The constant 2 should be dropped
The square should be on the 2
Show the slip
6 was not halved Half of 6 is 3, so the bracket is (x+3).
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Expand your completed-square answer in the margin. If it does not match the original, you have found your own error before the marker does.
If a question says 'hence' after completing the square, use your bracket — do not start again with the formula.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.