Revision Library

revisionlibrary.co.uk

KS3 · Algebra

KS3.M-A14 Introduction to Factorising Quadratics

AQA KS3.M-A14 · Calculators not allowed · about 75 minutes

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Factorise the quadratic expression fully: x² + 5x + 6

(1)

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

(3)

Factorise x² + 11x + 30 and hence find the values of x for which the expression equals zero.

(3)

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

(3)

Expand (x + 4)(x - 1) to form a quadratic. Then find all x such that this quadratic equals 6.

(4)

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)

Given f(x) = x² + 15x + 56, (a) factorise f(x) fully, (b) find f(-10).

(a)Factorise f(x) = x² + 15x + 56(2)

(b)Find f(-10).(2)

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)

Mark scheme · KS3.M-A14 Introduction to Factorising Quadratics

Question 1

B1 Correct factorisation (x + 2)(x + 3) cao
Answer: (x + 2)(x + 3)

Question 2

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

Question 3

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

Question 4

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 5

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 6

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

Question 7

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 8

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

Question 9

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 10

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 11

M1 Factorise correctly as (x + 5)(x + 6) or equivalent
A1 Set each factor to zero method shown
A1 Correct roots x = -5 and x = -6 cao
Answer: x = -5, x = -6

Question 12

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 13

M1 Correct expansion x² + 3x - 4
M1 Move 6 to make equation x² + 3x - 10 = 0
M1 Correct factorisation (x + 5)(x - 2) or equivalent
A1 Correct solutions x = -5 and x = 2 cao
Answer: x = -5, x = 2

Question 14

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 15

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 16

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 17

(a) M1 Identify two numbers multiply to 56 and add to 15
(a) A1 Correct factorisation (x + 7)(x + 8) cao
(a) Answer: (x + 7)(x + 8)
(b) M1 Substitute -10 into x² + 15x + 56 and evaluate method shown
(b) A1 Correct value f(-10) = 6 cao
(b) Answer: 6

Question 18

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

Introduction to Factorising Quadratics - Worksheets, Questions and Revision