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Grade 7 · Higher · free GCSE maths worksheet

Completing the square

Writing x^2 + bx + c as (x + p)^2 + q.

11 original questions, 40 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

Write x2 + 10x + 7 in the form (x + a)2 + b, where a and b are integers.

2 marks
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(x + 5)2 - 18
2

Write x2 + 8x + 3 in the form (x + a)2 + b, where a and b are integers.

2 marks
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(x + 4)2 - 13
3

  1. Write x2 − 5x + 1 in the form (x − a)2 + b.

  2. Hence write down the minimum value of x2 − 5x + 1 and the value of x for which it occurs.

3 marks
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a) (x - 5/2)2 - 21/4 b) minimum value -21/4 when x = 5/2
4

  1. Write x2 − 6x + 11 in the form (x − a)2 + b, where a and b are integers.

  2. Hence find the coordinates of the turning point of the curve y = x2 − 6x + 11

3 marks
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a) (x - 3)2 + 2 b) (3, 2)
5

The curve with equation y = x2 + bx + c has its turning point at (3, −4).
Find the values of b and c.

3 marks
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b) -6 c) 5
6

Solve x2 − 5x + 1 = 0 by completing the square.
Give your answers in the form p ± √q2 where p and q are integers.

4 marks
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x = (5 + √21)/2 or x = (5 - √21)/2
7

By completing the square, solve

x2 + 6x + 4 = 0

Give your answers in the form a ± √b where a and b are integers.

3 marks
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x = -3 + sqrt(5) or x = -3 - sqrt(5)
8

  1. Write 3 + 8x − x2 in the form a − (x − b)2, where a and b are integers.

  2. Hence write down the coordinates of the maximum point of the curve y = 3 + 8x − x2

4 marks
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a) 19 - (x - 4)2 b) (4, 19)
9

  1. Write 2x2 + 12x + 5 in the form a(x + b)2 + c, where a, b and c are integers.

  2. Hence write down the coordinates of the turning point of the curve y = 2x2 + 12x + 5

4 marks
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a) 2(x + 3)2 - 13 b) (-3, -13)
10

  1. Write 3x2 − 12x + 5 in the form a(x − b)2 + c, where a, b and c are integers.

  2. Hence solve 3x2 − 12x + 5 = 0
    Give your answers in the form p ± √qr where p, q and r are integers.

5 marks
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a) 3(x - 2)2 - 7 b) x = 2 ± √21/3
11

f(x) = 2x2 − 8x + 3

  1. Write f(x) in the form a(x − b)2 + c, where a, b and c are integers.

  2. Hence solve f(x) = 0, giving your answers in the form p ± √q2

  3. Priya says that the minimum point of the curve y = f(x) is (−2, −5).

    Explain the mistake Priya has made and write down the correct minimum point.

7 marks
Show answer
a) 2(x - 2)2 - 5 b) x = 2 + (√10)/2 or x = 2 - (√10)/2 c) The square is zero at x = 2 and its coefficient is positive, so Priya has the wrong sign; the minimum point is (2, -5)

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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