Robinson Tuition
Grade 8 · Higher · free GCSE maths worksheet

Quadratic inequalities

Solving and showing the solution set.

17 original questions, 66 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Solve x2 − 5x − 14 > 0

3 marks
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x 7
2

Solve x2 + 2x − 15 ≤ 0

Write your answer in set notation.

3 marks
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{x : -5 <= x <= 3}
3

Solve (x + 2)(x − 5) < 0

2 marks
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-2 < x < 5
4

Solve (x − 3)(x + 7) ≥ 0

2 marks
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x = 3
5

The graph of y = x2 − 3x − 10 is a U-shaped curve.

  1. Factorise x2 − 3x − 10 and write down where the curve crosses the x-axis.
  2. Use a sketch of the curve to solve x2 − 3x − 10 ≥ 0.
4 marks
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a) (x - 5)(x + 2); crosses at x = -2 and x = 5 b) x = 5
6

Solve x2 − 9x + 18 < 0

3 marks
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3 < x < 6
7

Solve x2 + 3x − 28 ≥ 0

3 marks
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x = 4
8

Solve x2 + 12 < 7x

4 marks
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3 < x < 4
9

n is an integer such that n2 − n − 6 < 0

Write down all the possible values of n.

3 marks
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-1, 0, 1, 2
10

n is an integer such that

n2 − 2n − 15 ≤ 0  and  3n + 1 > 4

Find all the possible values of n.

5 marks
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2, 3, 4, 5
11

Solve

5x − 4 > 2x + 8  and  x2 − 10x + 21 < 0

Give your answer as a single inequality.

5 marks
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4 < x < 7
12

Solve 3 < x2 − 6 ≤ 19

5 marks
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-5 <= x < -3 or 3 < x <= 5
13

Solve 2x2 - x - 15 < 0

4 marks
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-52 < x < 3
14

Solve 6x2 − 7x − 5 > 0

4 marks
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x 5/3
15

Solve 2x2 + 5x − 3 ≤ 0

5 marks
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-3 ≤ x ≤ 12
16

Solve

x2 − x − 20 ≤ 0  and  x2 − 4 > 0

5 marks
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-4 <= x < -2 or 2 < x <= 5
17

A rectangular allotment bed has length (x + 7) metres and width (x − 2) metres, where x > 2.

The area of the bed is less than 90 m2.

  1. Show that x2 + 5x − 104 < 0
  2. Find the range of possible values of x.

6 marks
Show answer
a) (x + 7)(x - 2) < 90 expands to x2 + 5x - 14 < 90, so x2 + 5x - 104 < 0 b) 2 < x < 8

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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