Completing the Square Questions with Answers

Free maths practice · Skills Lab

Completing the Square Questions with Answers

Here are 20 completing the square questions with answers, split into 10 levels that go from x² + 6x up to finding turning points and solving quadratic equations. For a fresh set of 10 questions at any level, open the same skill in our free Skills Lab, which also steps through the working one line at a time.

  • 10levels, easiest first
  • 20questions on this page, with answers
  • 10fresh questions every time in the Lab
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A worksheet shows the answer. The Skills Lab shows how to get there. Pick a level and the working appears one step at a time, each time you press the button, so your child can follow every move and then try ten questions of their own.

Skills Lab teach mode working through completing the square step by step
Teach: the working, one step at a timeA square diagram shows where the extra number comes from, which is why it has to be taken away again.
Skills Lab practise mode with ten completing the square questions
Practise: ten fresh questionsPress New 10 for a new set at the same level. Reveal each answer to check, or switch on Type the answer and let the Lab mark it.
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Practise completing the square in the Skills Lab

How to complete the square

Halve the number in front of x and put it in the bracket: x² + bx becomes (x + b/2)². Squaring that bracket adds an extra (b/2)², so take it away again, then tidy up the numbers. If there is a number in front of x², take it out as a factor first.

Write x² + 6x + 2 in the form (x + p)² + q

  1. Half of 6 is 3, so start with (x + 3)²
  2. (x + 3)² = x² + 6x + 9, which is 9 too many
  3. x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7

Find the turning point of y = x² + 6x + 2

  1. From above, y = (x + 3)² − 7
  2. The bracket is smallest (zero) when x = −3
  3. The turning point is (−3, −7), a minimum

Completing the Square Questions by level

Click a level to practise it
Level Question Question
Level 1No constant term 1. w² + 6w 2. r² + 8r
Level 2Add a constant 3. m² + 18m + 15 4. h² − 10h + 13
Level 3Odd linear term coefficient 5. x² − 3x 6. d² + 5d
Level 4Squared term coefficient 7. 3y² − 9y + 1 8. 2c² + 10c − 2
Level 5Negative squared term 9. 6 − 18h − 3h² 10. −f² − 8f + 3
Level 6Fractional squared term coefficient 11. (2/3)d² + 10d − 11 12. (1/2)k² − 2k − 2
Level 7Turning point 13. Turning point of y = x² − 2x + 3 14. Turning point of y = 4x − 2x² − 10
Level 8Solve 15. Solve 3f² + 12f − 3 = 0 16. Solve (1/2)c² + 6c + 15 = 0
Level 9Rearrange, then solve 17. Solve n(n + 11) = −1 18. Solve 2g² − 15g − 12 = g² − 9g + 5
Level 10Everything mixed 19. h² − 20h 20. −h² + 14h + 4
Show all 20 answers
Question Answer
1. w² + 6w (w + 3)² − 9
2. r² + 8r (r + 4)² − 16
3. m² + 18m + 15 (m + 9)² − 66
4. h² − 10h + 13 (h − 5)² − 12
5. x² − 3x (x − 3/2)² − 9/4
6. d² + 5d (d + 5/2)² − 25/4
7. 3y² − 9y + 1 3(y − 3/2)² − 23/4
8. 2c² + 10c − 2 2(c + 5/2)² − 29/2
9. 6 − 18h − 3h² −3(h + 3)² + 33
10. −f² − 8f + 3 −(f + 4)² + 19
11. (2/3)d² + 10d − 11 2/3(d + 15/2)² − 97/2
12. (1/2)k² − 2k − 2 1/2(k − 2)² − 4
13. Turning point of y = x² − 2x + 3 (1, 2) minimum
14. Turning point of y = 4x − 2x² − 10 (1, −8) maximum
15. Solve 3f² + 12f − 3 = 0 f = −2 ± √5
16. Solve (1/2)c² + 6c + 15 = 0 c = −6 ± √6
17. Solve n(n + 11) = −1 n = (−11 ± 3√13)/2
18. Solve 2g² − 15g − 12 = g² − 9g + 5 g = 3 ± √26
19. h² − 20h (h − 10)² − 100
20. −h² + 14h + 4 −(h − 7)² + 53

Common mistakes

  • Forgetting to take the square away. (x + 3)² is x² + 6x + 9, so you must subtract 9 to keep the expression the same.
  • Getting the sign of the turning point wrong. For (x + 3)² − 7 the turning point is at x = −3, not 3. The bracket is zero when x + 3 = 0.
  • Leaving the number in front of x² inside. For 2x² + 12x + 1, take out the 2 first: 2(x² + 6x) + 1, then complete the square inside.

Questions parents and students ask

What is completing the square used for?

It rewrites a quadratic so you can read off the turning point of its graph and solve equations that do not factorise.

Is completing the square on GCSE Foundation?

No. Completing the square is a GCSE Higher topic.

How do you find the turning point from completed square form?

For (x + p)² + q, the turning point is (−p, q). It is a minimum when the x² term is positive.

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