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5.7H Expanding and Factorising Quadratics: Higher Tier Practice

EDEXCEL Edexcel 1MA1 · Calculator allowed · about 70 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Expand and simplify fully: (2x + 3)(x - 5).

(Total for Question 1 is 1 mark)

Expand and simplify fully: (x + 4)(x + 1) - (x - 2)(x + 3).

(Total for Question 2 is 2 marks)

Factorise fully: 6x² - 13x - 5.

(Total for Question 3 is 3 marks)

Show that (x + 2)(x + k) expands to x² + (k + 2)x + 2k. Hence find the value of k if x² + 7x + 10 = (x + 2)(x + k).

(a)Show that (x + 2)(x + k) expands to x² + (k + 2)x + 2k.(1)

(b)Hence find the value of k.(3)

(Total for Question 4 is 4 marks)

The quadratic 4x² - 12x + p is a perfect square trinomial. Find p and write the quadratic as a square of a binomial.

(a)Find p so that 4x² - 12x + p is a perfect square trinomial.(2)

(b)Write 4x² - 12x + 9 as a square of a binomial.(2)

(Total for Question 5 is 4 marks)

Solve the quadratic equation by factorising: x² - x - 30 = 0. Give answers in exact form.

(Total for Question 6 is 5 marks)

Given that 3x² + bx - 16 = 0 has roots x = 2 and x = -8/3, find the value of b. Show working.

(Total for Question 7 is 5 marks)

Show that x² + (m - 3)x + (2 - m) is divisible by x - 1 for all values of m. Hence find the other (linear) factor.

(a)Show that x - 1 is a factor for all m by evaluating the expression at x = 1.(2)

(b)Hence find the other linear factor, and write down the full factorisation of the expression.(4)

(Total for Question 8 is 6 marks)

Solve the equation 2x² - 5x = 3 by completing the square. Give answers in exact form.

(Total for Question 9 is 6 marks)

Mark scheme · 5.7H Expanding and Factorising Quadratics: Higher Tier Practice

Question 1

B1 2x² -10x +3x -15 simplified to 2x² -7x -15 cao
Answer:2x² - 7x - 15

(Alternatively shown steps with correct simplification.)

Question 2

M1 expand both products correctly: (x²+5x+4) and (x² + x -6) or equivalent
A1 (x²+5x+4) - (x²+x-6) = x²+5x+4-x²-x+6 simplified to 4x + 10 cao
Answer: 4x + 10

Question 3

M1 find pair of numbers with product 6*(-5)=-30 and sum -13, e.g. -15 and 2, or equivalent method
M1 split middle term and factor by grouping to show (3x + 1)(2x - 5) or equivalent
A1 (3x + 1)(2x - 5) cao
Answer: (3x + 1)(2x - 5)

Question 4

(a) B1 expansion shown: x² + kx + 2x + 2k leading to x² + (k+2)x + 2k oe
(a) Answer: x² + (k + 2)x + 2k
(b) M1 equate coefficients: k + 2 = 7 and 2k = 10 or using either equation
(b) A1 k = 5 from k + 2 = 7 cao
(b) A1 consistency 2k = 10 gives k = 5 cao
(b) Answer: k = 5

Question 5

(a) M1 recognise form (ax)² - 2abx + b² or complete the square: (2x - m)² expands to 4x² -4mx + m²
(a) A1 match -4m = -12 so m = 3 then p = m² = 9 cao
(a) Answer: p = 9
(b) M1 show rewriting as (2x - 3)² or equivalent
(b) A1 (2x - 3)² cao
(b) Answer: (2x - 3)²

Question 6

M1 factorise to (x + 5)(x - 6) or show factors giving product -30 and sum -1
M1 set each factor to zero: x + 5 = 0 or x - 6 = 0
A1 x = -5
A1 x = 6
B1 list both solutions as x = -5, 6 (exact) cao
Answer:x = -5 or x = 6

(Exact values)

Question 7

M1 use factor form 3(x - r1)(x - r2) or substitute a root into equation
M1 construct factors from roots: factors proportional to (x - 2) and (x + 8/3) or clear fractions to 3(x - 2)(x + 8/3)
M1 expand 3(x - 2)(x + 8/3) correctly to get 3x² + bx -16 or equivalent
A1 show b = -6 + 8 = 2 or correct evaluation (see working)
B1 final answer b = 2 cao
Answer:b = 2

(Using 3(x - 2)(x + 8/3) -> expand: 3[x² + (8/3 - 2)x -16/3] = 3x² + 2x -16, so b = 2.)

Question 8

(a) M1 substitute x = 1 into the expression to obtain 1 + (m - 3) + (2 - m) or equivalent
(a) A1 simplify to 0, independent of m, so x = 1 is a root for every value of m and x - 1 is always a factor
(a) Answer: f(1) = 1 + (m - 3) + (2 - m) = 0 for every value of m, so x - 1 is always a factor.
(b) M1 attempt polynomial division (or write x² + (m-3)x + (2-m) = (x-1)(x+c) and compare coefficients)
(b) A1 correct method leading to quotient x + m - 2 (e.g. bring down terms of the division, or match coefficients c - 1 = m - 3)
(b) M1 check the constant term: -c = 2 - m gives c = m - 2, consistent with the x coefficient
(b) A1 full factorisation x² + (m - 3)x + (2 - m) = (x - 1)(x + m - 2) cao
(b) Answer: The other factor is x + m - 2. Full factorisation: x² + (m - 3)x + (2 - m) = (x - 1)(x + m - 2).

Question 9

M1 rearrange to 2x² -5x -3 = 0 or isolate quadratic and linear terms: 2x² -5x = 3 -> 2x² -5x -3 = 0
M1 divide by 2 to prepare for completing the square: x² - (5/2)x - 3/2 = 0
M1 complete the square: x² - (5/2)x + (25/16) = 3/2 + 25/16 or show adding (5/4)²
M1 write (x - 5/4)² = (24/16 + 25/16) = 49/16 or equivalent arithmetic
A1 take square roots to get x - 5/4 = ± 7/4
A1 x = 3 or x = -1/2 cao
Answer:x = 3 or x = -1/2

(Exact values)

Expanding and Factorising Quadratics: Higher Tier Practice - Worksheets, Questions and Revision