Quadratic Inequalities (GCSE Further Maths) - Worksheets, Questions and Revision

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GCSE · Algebra

A4 Quadratic Inequalities

AQA 8365 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Which of the following is the correct solution to the inequality x² - 16 ≤ 0 ?

A) x ≤ -4
B) x ≥ 4
C) -4 ≤ x ≤ 4
D) x ≤ -4 or x ≥ 4

(Total for Question 1 is 1 mark)

Solve the inequality x² - 9 > 0.

(Total for Question 2 is 2 marks)

Solve the inequality x² - 6x + 8 ≤ 0.

(Total for Question 3 is 3 marks)

Solve the inequality 2x² - 7x - 4 ≥ 0.

(Total for Question 4 is 3 marks)

Solve the inequality x² + 6 < 5x.

(Total for Question 5 is 3 marks)

The graph of y = x² + 2x - 8 is sketched below. Use the sketch to write down the solution to x² + 2x - 8 ≥ 0.

(Total for Question 6 is 2 marks)

Solve the inequality x² - 5x - 14 < 0. Give your answer using set notation.

(Total for Question 7 is 3 marks)

Solve the inequality x² - x - 6 ≤ 0. Hence write down all the integer values of x that satisfy the inequality.

(Total for Question 8 is 4 marks)

Solve the inequality 16 - x² < 0.

(Total for Question 9 is 3 marks)

Solve the inequality x² ≥ 49. Give your answer using set notation with the union symbol.

(Total for Question 10 is 3 marks)

Find the set of values of x that satisfy both x² - 9 < 0 and x + 1 > 0.

(Total for Question 11 is 4 marks)

A rectangular lawn has length (x + 3) metres and width (x - 1) metres, where x > 1. The area of the lawn is greater than 45 m². Form and solve an inequality to find the range of possible values of x.

(Total for Question 12 is 5 marks)

Solve the inequality x² + 4x - 3 > 0, giving your answer in surd form.

(Total for Question 13 is 4 marks)

Show that x² - 6x + 10 > 0 for all real values of x.

(Total for Question 14 is 3 marks)

Solve the inequality x² + 2x + 5 < 0. Explain what your answer tells you about the graph of y = x² + 2x + 5.

(Total for Question 15 is 3 marks)

Solve the inequality 12 - 4x - x² < 0.

(Total for Question 16 is 4 marks)

Solve the inequality x⁴ - 5x² + 4 < 0.

(Total for Question 17 is 6 marks)

The function f(x) = √20 - x - x² is defined for real values of x. Find the set of values of x for which f(x) is defined.

(Total for Question 18 is 4 marks)

The height, h metres, of a ball t seconds after being thrown is given by h = 1 + 15t - 5t². Find the values of t for which the ball is more than 11 metres above the ground.

(Total for Question 19 is 5 marks)

The equation x² + kx + (k + 3) = 0 has no real roots. Find the set of values of k.

(Total for Question 20 is 5 marks)

Mark scheme · A4 Quadratic Inequalities

Question 1

B1 correct option identified (C)
Answer: C) -4 ≤ x ≤ 4

Question 2

M1 factorise to (x-3)(x+3) and identify critical values x = -3 and x = 3
A1 correct inequality x < -3 or x > 3 (oe)
Answer: x < -3 or x > 3

Question 3

M1 factorise to (x-2)(x-4)
A1 correct critical values x = 2 and x = 4
A1 correct inequality 2 ≤ x ≤ 4 (oe)
Answer: 2 ≤ x ≤ 4

Question 4

M1 factorise to (2x+1)(x-4)
A1 correct critical values x = -1/2 and x = 4
A1 correct inequality x ≤ -1/2 or x ≥ 4 (oe)
Answer: x ≤ -1/2 or x ≥ 4

Question 5

M1 rearrange to x² - 5x + 6 < 0
M1 factorise to (x-2)(x-3)
A1 correct inequality 2 < x < 3 (oe)
Answer: 2 < x < 3

Question 6

B1 correctly reads roots x = -4 and x = 2 from the sketch
B1 correct inequality x ≤ -4 or x ≥ 2 (oe)
Answer: x ≤ -4 or x ≥ 2

Question 7

M1 factorise to (x-7)(x+2)
A1 correct critical values x = -2 and x = 7
A1 correct set notation {x : -2 < x < 7} (oe)
Answer: {x : -2 < x < 7}

Question 8

M1 factorise to (x-3)(x+2)
A1 correct critical values x = -2 and x = 3
A1 correct inequality -2 ≤ x ≤ 3 (oe)
B1 correct list of integers -2, -1, 0, 1, 2, 3 (ft from their inequality)
Answer: -2 ≤ x ≤ 3; integers -2, -1, 0, 1, 2, 3

Question 9

M1 rearrange to x² > 16, correctly reversing the inequality
A1 correct critical values x = -4 and x = 4
A1 correct inequality x < -4 or x > 4 (oe)
Answer: x < -4 or x > 4

Question 10

M1 identify critical values x = -7 and x = 7
A1 correct inequality x ≤ -7 or x ≥ 7 (oe)
B1 correct set notation {x : x ≤ -7} U {x : x ≥ 7} (oe)
Answer: {x : x ≤ -7} U {x : x ≥ 7}

Question 11

M1 solve x²-9<0 to obtain -3<x<3
A1 correct -3 < x < 3
M1 solve x+1>0 to obtain x>-1
A1 correct combined interval -1 < x < 3 (ft)
Answer: -1 < x < 3

Question 12

M1 form area expression (x+3)(x-1)
M1 form inequality x²+2x-3>45 or equivalent x²+2x-48>0
M1 factorise to (x+8)(x-6)
A1 correct critical values x = -8 and x = 6
A1 correct final answer x > 6, rejecting x < -8 as it contradicts x > 1 (cao)
Answer: x > 6

Question 13

M1 apply the quadratic formula with a=1, b=4, c=-3
A1 correct roots x = -2 - √7 and x = -2 + √7 simplified
M1 identify correct inequality structure (outside the roots) using sign reasoning
A1 fully correct answer x < -2 - √7 or x > -2 + √7 (oe)
Answer: x < -2 - √7 or x > -2 + √7

Question 14

M1 attempt to complete the square
A1 correct form (x-3)² + 1
A1 cso: since (x-3)² ≥ 0 for all real x, (x-3)²+1 ≥ 1 > 0 for all real x
Answer: x² - 6x + 10 = (x-3)² + 1 > 0 for all real x

Question 15

M1 attempt discriminant b²-4ac = 4-20 = -16, or complete the square to (x+1)²+4
A1 correct conclusion: no real solutions (discriminant negative / minimum value 4 > 0)
B1 correct interpretation: the graph of y=x²+2x+5 never crosses or touches the x-axis (lies entirely above it)
Answer: No real solutions; the graph lies entirely above the x-axis

Question 16

M1 rearrange (e.g. multiply by -1, reversing the inequality) to x²+4x-12>0
M1 factorise to (x+6)(x-2)
A1 correct critical values x = -6 and x = 2
A1 correct inequality x < -6 or x > 2 (oe)
Answer: x < -6 or x > 2

Question 17

M1 substitute u=x² to obtain u²-5u+4<0
M1 factorise to (u-1)(u-4)<0
A1 correct critical values u = 1 and u = 4, giving 1 < u < 4
M1 convert to 1 < x² < 4, splitting into x²>1 and x²<4
A1 correct critical x values -2, -1, 1, 2
A1 fully correct final answer -2 < x < -1 or 1 < x < 2 (oe)
Answer: -2 < x < -1 or 1 < x < 2

Question 18

M1 set up condition 20-x-x² ≥ 0 for the square root to be defined
M1 rearrange (e.g. multiply by -1) and factorise to (x+5)(x-4) ≤ 0
A1 correct critical values x = -5 and x = 4
A1 correct final domain -5 ≤ x ≤ 4 (oe)
Answer: -5 ≤ x ≤ 4

Question 19

M1 form inequality 1+15t-5t²>11
M1 rearrange and simplify (e.g. divide by -5, reversing) to t²-3t+2<0
M1 factorise to (t-1)(t-2)
A1 correct critical values t = 1 and t = 2
A1 correct final answer 1 < t < 2 (seconds) (oe)
Answer: 1 < t < 2 (seconds)

Question 20

M1 write discriminant b²-4ac = k²-4(k+3)
M1 simplify to k²-4k-12 (<0)
M1 factorise to (k-6)(k+2)
A1 correct critical values k = -2 and k = 6
A1 correct final answer -2 < k < 6 (oe)
Answer: -2 < k < 6

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Quadratic Inequalities (GCSE Further Maths) - Worksheets, Questions and Revision

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