Solving Quadratics Worksheet with Answers

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GCSE · Algebra

5.8 Solving Quadratics

EDEXCEL 1MA1 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

Solve x² + 5x + 6 = 0

(a)Solve x² + 5x + 6 = 0(3)

(Total for Question 1 is 3 marks)

2

Solve x² - 49 = 0

(a)Solve x² - 49 = 0(2)

(Total for Question 2 is 2 marks)

3

Solve 4x² - 25 = 0, giving your answers as exact values.

(a)Solve 4x² - 25 = 0, giving your answers as exact values(2)

(Total for Question 3 is 2 marks)

4

Solve x² + 4x = 0

(a)Solve x² + 4x = 0(2)

(Total for Question 4 is 2 marks)

5

Which value of x is a solution to x² - 3x - 10 = 0?

(a)Circle the correct answer.(1)

A) x = 5
B) x = 4
C) x = -4
D) x = 3

(Total for Question 5 is 1 mark)

6

Solve x² = 7x

(a)Solve x² = 7x(2)

(Total for Question 6 is 2 marks)

7

Solve 2x² + 7x + 3 = 0

(a)Solve 2x² + 7x + 3 = 0(3)

(Total for Question 7 is 3 marks)

8

Solve 3x² - 11x - 4 = 0

(a)Solve 3x² - 11x - 4 = 0(3)

(Total for Question 8 is 3 marks)

9

Solve x² - 6x + 4 = 0. Use the quadratic formula and give your answers correct to 2 decimal places.

(a)Solve x² - 6x + 4 = 0, giving your answers correct to 2 decimal places(3)

(Total for Question 9 is 3 marks)

10

Solve 3x² - 5x - 1 = 0. Use the quadratic formula and give your answers correct to 3 significant figures.

(a)Solve 3x² - 5x - 1 = 0, giving your answers correct to 3 significant figures(3)

(Total for Question 10 is 3 marks)

11

A rectangle has length (x + 5) cm and width x cm. The area of the rectangle is 84 cm².

Diagram NOT accurately drawn

(a)Show that x² + 5x - 84 = 0(2)

(b)Solve the equation to find the value of x, given that x > 0(3)

(c)Hence find the length of the rectangle(1)

(Total for Question 11 is 6 marks)

12

The graph of y = x² - 2x - 15 crosses the x-axis at the points A and B.

(a)Find the coordinates of A and B(3)

(Total for Question 12 is 3 marks)

13

Two consecutive positive integers have a product of 132. Let n be the smaller integer.

(a)Show that n² + n - 132 = 0(2)

(b)Solve the equation to find the two consecutive integers(3)

(Total for Question 13 is 5 marks)

14

Solve (x + 2)(x - 1) = 18

(a)Solve (x + 2)(x - 1) = 18(4)

(Total for Question 14 is 4 marks)

15

f(x) = x² - 6x + 11

(a)Express f(x) in the form (x - a)² + b, where a and b are integers(2)

(b)Hence write down the minimum value of f(x) and the value of x at which it occurs(2)

(Total for Question 15 is 4 marks)

16

Solve x² + 8x + 3 = 0 by completing the square. Leave your answer in surd form, giving each surd as simply as possible.

(a)Solve x² + 8x + 3 = 0 by completing the square, leaving your answer in surd form(3)

(Total for Question 16 is 3 marks)

17

Solve x² - 4x - 7 = 0. Give your answers in the form a ± √b, where a and b are integers.

(a)Solve x² - 4x - 7 = 0, giving your answers in the form a ± √b(3)

(Total for Question 17 is 3 marks)

18

A ball is thrown into the air. Its height above the ground, h metres, after t seconds is given by h = 20t - 5t²

(a)Find the values of t when the ball is at ground level(3)

(b)Find the times at which the height of the ball is 15 metres(4)

(Total for Question 18 is 7 marks)

19

The diagram shows a right-angled triangle with sides x cm, (x + 7) cm and (x + 8) cm, where (x + 8) cm is the hypotenuse.

Diagram NOT accurately drawn

(a)Show that x² - 2x - 15 = 0(3)

(b)Solve the equation to find x, and hence find the perimeter of the triangle(4)

(Total for Question 19 is 7 marks)

Mark scheme · 5.8 Solving Quadratics

Question 1

(a) M1 correctly factorises to (x + 2)(x + 3) = 0
(a) A1 x = -2
(a) A1 x = -3 (both values required, ft from correct factorisation)
(a) Answer: x = -2 or x = -3

Question 2

(a) M1 rearranges to x² = 49 oe, or factorises to (x - 7)(x + 7) = 0
(a) A1 x = 7 and x = -7 (both values required, cao)
(a) Answer: x = 7 or x = -7

Question 3

(a) M1 rearranges to x² = 25/4 oe, or factorises to (2x - 5)(2x + 5) = 0
(a) A1 x = 2.5 and x = -2.5 oe (both values required, cao)
(a) Answer: x = 2.5 or x = -2.5

Question 4

(a) M1 factorises to x(x + 4) = 0
(a) A1 x = 0 and x = -4 (both values required, cao)
(a) Answer: x = 0 or x = -4

Question 5

(a) B1 A cao
(a) Answer: A

Question 6

(a) M1 rearranges to x² - 7x = 0 and factorises to x(x - 7) = 0 (dividing both sides by x loses a solution and does not score this mark)
(a) A1 x = 0 and x = 7 (both values required, cao)
(a) Answer: x = 0 or x = 7

Question 7

(a) M1 correctly factorises to (2x + 1)(x + 3) = 0
(a) A1 x = -3
(a) A1 x = -0.5 oe (both values required, ft from correct factorisation)
(a) Answer: x = -3 or x = -0.5

Question 8

(a) M1 correctly factorises to (3x + 1)(x - 4) = 0
(a) A1 x = 4
(a) A1 x = -1/3 oe (both values required, ft from correct factorisation)
(a) Answer: x = 4 or x = -1/3

Question 9

(a) M1 correct substitution into the quadratic formula, x = (6 ± √36 - 16) / 2
(a) A1 x = 5.24 awrt
(a) A1 x = 0.76 awrt (both values required)
(a) Answer: x = 5.24 or x = 0.76 (2dp)

Question 10

(a) M1 correct substitution into the quadratic formula, x = (5 ± √25 + 12) / 6
(a) A1 x = 1.85 awrt
(a) A1 x = -0.180 awrt (both values required)
(a) Answer: x = 1.85 or x = -0.180 (3sf)

Question 11

(a) M1 forms the equation x(x + 5) = 84
(a) A1 cso, rearranges correctly to x² + 5x - 84 = 0
(a) Answer: x² + 5x - 84 = 0 (shown)
(b) M1 correctly factorises to (x + 12)(x - 7) = 0
(b) A1 identifies both roots x = 7 and x = -12
(b) A1 (dep) selects x = 7, rejecting x = -12 since x > 0
(b) Answer: x = 7
(c) B1 12 cm, ft from x = 7 in part (b)
(c) Answer: 12 cm

Question 12

(a) M1 correctly factorises to (x - 5)(x + 3) = 0, or sets y = 0 and uses the quadratic formula
(a) A1 (-3, 0)
(a) A1 (5, 0) (both coordinate pairs required, either order)
(a) Answer: A = (-3, 0), B = (5, 0)

Question 13

(a) M1 forms the equation n(n + 1) = 132
(a) A1 cso, rearranges correctly to n² + n - 132 = 0
(a) Answer: n² + n - 132 = 0 (shown)
(b) M1 correctly factorises to (n - 11)(n + 12) = 0
(b) A1 (dep) n = 11, rejecting n = -12 since the integers are positive
(b) A1 states the integers are 11 and 12
(b) Answer: 11 and 12

Question 14

(a) M1 expands and rearranges to x² + x - 20 = 0 (or equivalent three-term quadratic equal to zero)
(a) M1 correctly factorises to (x + 5)(x - 4) = 0
(a) A1 x = -5
(a) A1 x = 4 (both values required)
(a) Answer: x = -5 or x = 4

Question 15

(a) M1 (x - 3)² seen
(a) A1 (x - 3)² + 2 cao
(a) Answer: (x - 3)² + 2
(b) B1 minimum value = 2, ft from part (a)
(b) B1 occurs at x = 3, ft from part (a)
(b) Answer: minimum value 2 at x = 3

Question 16

(a) M1 correctly completes the square, (x + 4)² - 13 = 0
(a) M1 rearranges to x + 4 = ± √13
(a) A1 x = -4 + √13 or x = -4 - √13 oe (both terms required, cao)
(a) Answer: x = -4 + √13 or x = -4 - √13

Question 17

(a) M1 correctly completes the square, (x - 2)² - 11 = 0, or correct substitution into the quadratic formula
(a) M1 rearranges to x - 2 = ± √11 (or equivalent simplification of √44/2)
(a) A1 x = 2 ± √11 cao
(a) Answer: x = 2 + √11 or x = 2 - √11

Question 18

(a) M1 sets 20t - 5t² = 0 and factorises to 5t(4 - t) = 0
(a) A1 t = 0
(a) A1 t = 4 (both values required)
(a) Answer: t = 0 or t = 4
(b) M1 sets 20t - 5t² = 15 and rearranges to t² - 4t + 3 = 0 (or equivalent unsimplified form)
(b) M1 correctly factorises to (t - 1)(t - 3) = 0
(b) A1 t = 1
(b) A1 t = 3 (both values required)
(b) Answer: t = 1 or t = 3

Question 19

(a) M1 uses Pythagoras' theorem to form the equation (x + 7)² + x² = (x + 8)²
(a) M1 correctly expands all three terms, x² + 14x + 49 + x² = x² + 16x + 64
(a) A1 cso, rearranges correctly to x² - 2x - 15 = 0
(a) Answer: x² - 2x - 15 = 0 (shown)
(b) M1 correctly factorises to (x - 5)(x + 3) = 0
(b) A1 (dep) selects x = 5, rejecting x = -3 since a length cannot be negative
(b) M1 finds all three side lengths, 5 cm, 12 cm and 13 cm
(b) A1 perimeter = 30 cm cao
(b) Answer: x = 5, perimeter = 30 cm

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5.8 Solving Quadratics

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Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

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Question 19