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GCSE · Algebra

5.7 Expanding and Factorising Quadratics

EDEXCEL 1MA1 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Expand each expression.

(a)Expand 3(x + 5)(1)

(b)Expand 4(2x - 3)(1)

(Total for Question 1 is 2 marks)

Expand and simplify each expression.

(a)Expand and simplify 5(x + 2) + 3(x - 4)(2)

(b)Expand and simplify 6(2x - 1) - 2(x - 5)(2)

(Total for Question 2 is 4 marks)

Expand and simplify each expression.

(a)(x + 2)(x + 9)(2)

(b)(x + 6)(x + 6)(2)

(Total for Question 3 is 4 marks)

Which of these is the correct expansion of (x + 4)(x - 6)?

(a)Circle the correct answer.(1)

A) x² - 2x - 24
B) x² + 2x - 24
C) x² - 10x - 24
D) x² - 2x + 24

(Total for Question 4 is 1 mark)

Expand and simplify each expression.

(a)(x - 3)(x + 8)(2)

(b)(x - 5)(x - 9)(2)

(c)(2x + 3)(x - 4)(2)

(Total for Question 5 is 6 marks)

Expand and simplify each expression.

(a)(x + 7)(x - 7)(2)

(b)(x - 6)(x - 6)(3)

(Total for Question 6 is 5 marks)

Factorise each expression.

(a)Factorise x² + 9x + 20(2)

(b)Factorise x² + 2x - 15(2)

(Total for Question 7 is 4 marks)

Factorise each expression.

(a)Factorise x² - 11x + 28(2)

(b)Factorise x² - 6x - 27(2)

(Total for Question 8 is 4 marks)

Factorise each expression fully.

(a)Factorise fully 2x² + 8x - 24(3)

(b)Factorise fully 3x² - 27(3)

(Total for Question 9 is 6 marks)

Factorise each expression.

(a)Factorise x² - 64(2)

(b)Factorise x² - 121(2)

(Total for Question 10 is 4 marks)

Show that (x + 3)(x - 5) + 4 = x² - 2x - 11

(a)Show that (x + 3)(x - 5) + 4 = x² - 2x - 11(3)

(Total for Question 11 is 3 marks)

Solve x² + 3x - 10 = 0

(a)Solve x² + 3x - 10 = 0(3)

(Total for Question 12 is 3 marks)

Solve x² + 4x = 21

(a)Solve x² + 4x = 21(4)

(Total for Question 13 is 4 marks)

A rectangle has length (x + 2) cm and width (x - 1) cm. The area of the rectangle is 40 cm^2.

Diagram NOT accurately drawn

(a)Show that x² + x - 42 = 0(3)

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

(Total for Question 14 is 6 marks)

Prove algebraically that (n + 1)² - (n - 1)² is always a multiple of 4, for any positive integer n.

(a)Prove algebraically that (n + 1)² - (n - 1)² is always a multiple of 4(4)

(Total for Question 15 is 4 marks)

Factorise each expression.

(a)Factorise 2x² + 11x + 12(3)

(b)Factorise 3x² + 11x + 6(3)

(Total for Question 16 is 6 marks)

Simplify fully (x² - 9) / (x² + 5x + 6)

(a)Simplify fully (x² - 9) / (x² + 5x + 6)(3)

(Total for Question 17 is 3 marks)

A square has side length (x + 3) cm. The area of the square is equal to the area of a rectangle with length (2x + 1) cm and width 5 cm.

Diagram NOT accurately drawn

(a)Work out the value of x(4)

(Total for Question 18 is 4 marks)

Mark scheme · 5.7 Expanding and Factorising Quadratics

Question 1

(a) B1 3x + 15 oe
(a) Answer: 3x + 15
(b) B1 8x - 12 oe
(b) Answer: 8x - 12

Question 2

(a) M1 correct expansion of at least one bracket, 5x + 10 or 3x - 12
(a) A1 8x - 2 cao
(a) Answer: 8x - 2
(b) M1 correct expansion of at least one bracket, 12x - 6 or -2x + 10
(b) A1 10x + 4 cao
(b) Answer: 10x + 4

Question 3

(a) M1 at least 2 of x², 11x (or 9x + 2x), 18 correct
(a) A1 x² + 11x + 18 cao
(a) Answer: x² + 11x + 18
(b) M1 at least 3 correct terms from x², 6x, 6x, 36
(b) A1 x² + 12x + 36 cao
(b) Answer: x² + 12x + 36

Question 4

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

Question 5

(a) M1 at least 3 of x², 8x, -3x, -24 correct
(a) A1 x² + 5x - 24 cao
(a) Answer: x² + 5x - 24
(b) M1 at least 3 of x², -9x, -5x, 45 correct
(b) A1 x² - 14x + 45 cao
(b) Answer: x² - 14x + 45
(c) M1 at least 3 of 2x², -8x, 3x, -12 correct
(c) A1 2x² - 5x - 12 cao
(c) Answer: 2x² - 5x - 12

Question 6

(a) M1 x² - 7x + 7x - 49 or equivalent, showing middle terms
(a) A1 x² - 49 cao
(a) Answer: x² - 49
(b) M1 attempt at full expansion, x² - 6x - 6x + 36 (at least 3 terms correct)
(b) M1 correctly collects -6x - 6x = -12x
(b) A1 x² - 12x + 36 cao
(b) Answer: x² - 12x + 36

Question 7

(a) M1 correct factor pair of 20 that sums to 9 identified (4 and 5), or (x+4)(x+5) seen
(a) A1 (x + 4)(x + 5) cao
(a) Answer: (x + 4)(x + 5)
(b) M1 correct factor pair of -15 that sums to 2 identified (5 and -3)
(b) A1 (x + 5)(x - 3) cao
(b) Answer: (x + 5)(x - 3)

Question 8

(a) M1 correct factor pair of 28 that sums to -11 identified (-4 and -7)
(a) A1 (x - 4)(x - 7) cao
(a) Answer: (x - 4)(x - 7)
(b) M1 correct factor pair of -27 that sums to -6 identified (-9 and 3)
(b) A1 (x - 9)(x + 3) cao
(b) Answer: (x - 9)(x + 3)

Question 9

(a) M1 takes out common factor of 2, 2(x² + 4x - 12)
(a) M1 correctly factorises the bracket, (x + 6)(x - 2)
(a) A1 2(x + 6)(x - 2) cao
(a) Answer: 2(x + 6)(x - 2)
(b) M1 takes out common factor of 3, 3(x² - 9)
(b) M1 recognises difference of two squares, (x + 3)(x - 3)
(b) A1 3(x + 3)(x - 3) cao
(b) Answer: 3(x + 3)(x - 3)

Question 10

(a) M1 recognises √64 = 8
(a) A1 (x + 8)(x - 8) cao
(a) Answer: (x + 8)(x - 8)
(b) M1 recognises √121 = 11
(b) A1 (x + 11)(x - 11) cao
(b) Answer: (x + 11)(x - 11)

Question 11

(a) M1 expands (x+3)(x-5) to give x² - 5x + 3x - 15 (or equivalent unsimplified form)
(a) M1 (dep) adds 4 to reach x² - 2x - 11
(a) A1 cso, fully correct working shown leading to the printed answer with no errors
(a) Answer: x² - 2x - 11 (shown)

Question 12

(a) M1 correctly factorises to (x + 5)(x - 2) = 0
(a) A1 x = 2
(a) A1 x = -5 (both values required, ft from correct factorisation)
(a) Answer: x = 2 or x = -5

Question 13

(a) M1 rearranges to x² + 4x - 21 = 0
(a) M1 correctly factorises to (x + 7)(x - 3) = 0
(a) A1 x = 3
(a) A1 x = -7 (both values required)
(a) Answer: x = 3 or x = -7

Question 14

(a) M1 forms the equation (x + 2)(x - 1) = 40
(a) M1 expands the left-hand side to x² + x - 2 = 40 (or equivalent)
(a) A1 cso, rearranges correctly to x² + x - 42 = 0
(a) Answer: x² + x - 42 = 0 (shown)
(b) M1 correctly factorises to (x + 7)(x - 6) = 0
(b) A1 identifies both roots x = 6 and x = -7
(b) A1 (dep) selects x = 6, rejecting x = -7 since x > 0
(b) Answer: x = 6

Question 15

(a) M1 expands (n + 1)² = n² + 2n + 1
(a) M1 expands (n - 1)² = n² - 2n + 1
(a) M1 (dep) subtracts to give (n²+2n+1) - (n²-2n+1) = 4n
(a) A1 cso, correct conclusion that 4n is a multiple of 4 since n is an integer
(a) Answer: (n+1)² - (n-1)² = 4n, which is a multiple of 4 for any integer n

Question 16

(a) M1 attempts (2x + p)(x + q) with pq = 12 tried
(a) M1 correct combination found, cross terms give 8x + 3x = 11x
(a) A1 (2x + 3)(x + 4) cao
(a) Answer: (2x + 3)(x + 4)
(b) M1 attempts (3x + p)(x + q) with pq = 6 tried
(b) M1 correct combination found, cross terms give 9x + 2x = 11x
(b) A1 (3x + 2)(x + 3) cao
(b) Answer: (3x + 2)(x + 3)

Question 17

(a) M1 factorises numerator, (x + 3)(x - 3)
(a) M1 factorises denominator, (x + 2)(x + 3)
(a) A1 (x - 3) / (x + 2) cao, after cancelling common factor (x + 3)
(a) Answer: (x - 3) / (x + 2)

Question 18

(a) M1 forms the equation (x + 3)² = 5(2x + 1)
(a) M1 expands and rearranges to a three-term quadratic equal to zero, x² - 4x + 4 = 0 (or equivalent)
(a) M1 correctly factorises to (x - 2)(x - 2) = 0
(a) A1 x = 2 cao
(a) Answer: x = 2

Expanding and Factorising Quadratics - Worksheets, Questions and Revision

5.7 Expanding and Factorising Quadratics

Question 1

Question 2

Question 3

Question 4

Question 5

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