Introduction to Factorising Quadratics - Worksheets, Questions and Revision

29 original exam-style questions - 6 pages of questions with a full mark scheme - free printable PDF.

Download PDF Jump to mark scheme (page 7)Introduction to Factorising Quadratics revision guide

This topic is chapter 1 of KS3 Maths: Algebra Practice Book 2.

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2.14 Introduction to Factorising Quadratics

AQA KS3.M-A14 · Calculators not allowed · about 90 minutes

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Factorise x² + 6x + 5 and use the factorised form to solve x² + 6x + 5 = 0

(3)

Factorise fully and solve x² - 5x - 24 = 0

(3)

Factorise x² + 13x + 36 fully and state the roots of the quadratic equation x² + 13x + 36 = 0.

(4)

The product of two consecutive integers is 56. Let the smaller integer be x. (a) Write an equation in x. (b) Solve the equation to find the positive integer.

(5)

Factorise fully: 9x² - 25

(3)

Show that x² + 6x + 9 can be written as (x + 3)². Use this to explain why x² + 6x + 9 is never negative for any real x. Give the value(s) of x for which the expression equals 0.

(6)

Factorise: x² + 2x - 8

(1)

Factorise: x² - 7x + 12

(1)

Factorise fully: x² + 9x + 20

(2)

Factorise fully: x² - x - 6

(2)

Given that (x + 3)(x + 4) = x² + bx + c, state the values of b and c.

(2)

Multiply out (x + 5)(x - 2) and then factorise your answer to show you get back the original factors.

(2)

Factorise the difference of two squares: x² - 9

(2)

Factorise fully: 4x² - 36

(2)

Factorise the quadratic expression fully: x² + 7x + 10

(1)

Factorise x² + 14x + 45 and hence find the values of x for which the expression equals zero.

(3)

Factorise fully: 25x² - 64

(1)

Expand (x + 6)(x - 2) to form a quadratic. Then find all values of x such that this quadratic equals 9.

(3)

Factorise x² + 17x + 72 fully and state the roots of the quadratic equation x² + 17x + 72 = 0.

(3)

Given that f(x) = x² + 19x + 90:
(a) factorise f(x) fully
(b) find f(-12).

(3)

Show that x² + 10x + 25 can be written as (x + 5)². Use this to explain why x² + 10x + 25 is never negative for any real x. Give the value(s) of x for which the expression equals 0.

(3)

Factorise fully: 4x² - 49. Hence solve 4x² - 49 = 0.

(3)

Factorise: x² + 3x - 10

(1)

Factorise: x² - 9x + 18

(1)

Factorise fully: x² + 11x + 24

(2)

Factorise fully: x² - 2x - 15

(2)

Factorise the difference of two squares: x² - 16

(1)

Factorise fully: 9x² - 100

(2)

Factorise x² + 8x + 7 and use the factorised form to solve x² + 8x + 7 = 0

(3)

Mark scheme · 2.14 Introduction to Factorising Quadratics

Question 1

M1 Correct factorisation (x + 1)(x + 5) or equivalent
A1 Set each factor to zero: x + 1 = 0, x + 5 = 0
A1 Correct solutions x = -1 and x = -5 cao
Answer: x = -1, x = -5

Question 2

M1 Factorise correctly to (x - 8)(x + 3) or equivalent
A1 Set factors to zero shown (x - 8 = 0, x + 3 = 0)
A1 Correct solutions x = 8 and x = -3 cao
Answer: x = 8, x = -3

Question 3

M1 Identify two numbers multiply to 36 and add to 13
M1 Correct factorisation (x + 4)(x + 9)
A1 Set factors to zero method shown
A1 Correct roots x = -4 and x = -9 cao
Answer: x = -4, x = -9

Question 4

M1 Correct equation x(x + 1) = 56 or equivalent shown
M1 Rearrange to x² + x - 56 = 0
M1 Correct factorisation (x + 8)(x - 7) or equivalent
A1 Correct solutions x = 7 and x = -8
A1 State positive integer is 7 cao
Answer: 7

Question 5

M1 Recognise as difference of squares 9x² - 25 = (3x)² - 5²
A1 Correct factorisation (3x - 5)(3x + 5) cao
B1 Both linear factors given in correct order or equivalent
Answer: (3x - 5)(3x + 5)

Question 6

M1 Complete the square or expand (x + 3)² showing x² + 6x + 9
M1 State the identity x² + 6x + 9 = (x + 3)² explicitly
A1 Explain that a square is always ≥ 0, so (x + 3)² ≥ 0 for all real x
M1 Conclude that x² + 6x + 9 is never negative by following from the square property
A1 State that the minimum value is 0 and it occurs when x + 3 = 0
A1 Give the value x = -3 for which the expression equals 0 cao
Answer: x = -3

Question 7

B1 Correct factorisation (x + 4)(x - 2) cao
Answer: (x + 4)(x - 2)

Question 8

B1 Correct factorisation (x - 3)(x - 4) cao
Answer: (x - 3)(x - 4)

Question 9

M1 Identify two numbers multiply to 20 and add to 9
A1 Correct factorisation (x + 4)(x + 5) cao
Answer: (x + 4)(x + 5)

Question 10

M1 Find two numbers multiply to -6 and add to -1
A1 Correct factorisation (x - 3)(x + 2) cao
Answer: (x - 3)(x + 2)

Question 11

M1 Multiply out to get x² + 7x + 12 or equivalent
A1 b = 7 and c = 12 cao
Answer: b = 7, c = 12

Question 12

M1 Correct expansion x² + 3x - 10
A1 Correct factorisation back to (x + 5)(x - 2) cao
Answer: Expansion: x² + 3x - 10; Factorised: (x + 5)(x - 2)

Question 13

M1 Recognise 9 = 3² and use difference of squares method
A1 Correct factorisation (x - 3)(x + 3) cao
Answer: (x - 3)(x + 3)

Question 14

M1 Factor out common factor 4 to get 4(x² - 9)
A1 Complete factorisation 4(x - 3)(x + 3) cao
Answer: 4(x - 3)(x + 3)

Question 15

B1 (x + 2)(x + 5) cao
Answer: (x + 2)(x + 5)

Question 16

M1 correct pair of factors of 45 that sum to 14 identified
A1 (x + 5)(x + 9) cao
B1 x = -5 and x = -9
Answer: (x + 5)(x + 9); x = -5 or x = -9

Question 17

B1 (5x - 8)(5x + 8) cao
Answer: (5x - 8)(5x + 8)

Question 18

M1 expands to x² + 4x - 12
M1 sets up and factorises x² + 4x - 21 = 0 as (x + 7)(x - 3) = 0
A1 x = -7 and x = 3
Answer: x² + 4x - 12; x = -7 or x = 3

Question 19

M1 correct pair of factors of 72 that sum to 17 identified
A1 (x + 8)(x + 9) cao
B1 x = -8 and x = -9
Answer: (x + 8)(x + 9); x = -8 or x = -9

Question 20

M1 correct pair of factors of 90 that sum to 19 identified
A1 (x + 9)(x + 10) cao
B1 f(-12) = 6 cao (ft their factorised form or by direct substitution)
Answer: (x + 9)(x + 10); f(-12) = 6

Question 21

M1 expands (x + 5)² to confirm x² + 10x + 25
A1 explains a square is never negative, so the expression is never negative
B1 x = -5
Answer: x² + 10x + 25 = (x + 5)², never negative since a square cannot be negative; equals 0 only when x = -5

Question 22

M1 identifies 4x² = (2x)² and 49 = 7²
A1 (2x - 7)(2x + 7) cao
B1 x = 3.5 and x = -3.5
Answer: (2x - 7)(2x + 7); x = 3.5 or x = -3.5

Question 23

B1 (x - 2)(x + 5) cao
Answer: (x - 2)(x + 5)

Question 24

B1 (x - 6)(x - 3) cao
Answer: (x - 6)(x - 3)

Question 25

M1 correct pair of factors of 24 that sum to 11 identified
A1 (x + 3)(x + 8) cao
Answer: (x + 3)(x + 8)

Question 26

M1 correct pair of factors of -15 that sum to -2 identified
A1 (x - 5)(x + 3) cao
Answer: (x - 5)(x + 3)

Question 27

B1 (x - 4)(x + 4) cao
Answer: (x - 4)(x + 4)

Question 28

M1 identifies 9x² = (3x)² and 100 = 10²
A1 (3x - 10)(3x + 10) cao
Answer: (3x - 10)(3x + 10)

Question 29

M1 correct pair of factors of 7 that sum to 8 identified
A1 (x + 1)(x + 7) cao
B1 x = -1 and x = -7
Answer: (x + 1)(x + 7); x = -1 or x = -7

Introduction to Factorising Quadratics - Worksheets, Questions and Revision

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2.14 Introduction to Factorising Quadratics

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