Which of the following is the correct factorisation of x2 + 9x + 20?
A) (x+4)(x+5)
B) (x+2)(x+10)
C) (x-4)(x-5)
D) (x+20)(x+1)
(Total for Question 1 is 1 mark)
2
Factorise x2 - 3x - 40
(Total for Question 2 is 1 mark)
3
Factorise 4x2 - 25
(Total for Question 3 is 1 mark)
4
Factorise fully 3x2 - 27
(Total for Question 4 is 1 mark)
5
Solve the equation x2 + 2x - 24 = 0 by factorising.
(Total for Question 5 is 1 mark)
6
Solve the equation x2 - 11x + 30 = 0 by factorising.
(Total for Question 6 is 1 mark)
7
Write down the coordinates of the minimum point of the curve y = (x - 3)2 + 7.
(Total for Question 7 is 1 mark)
8
Factorise fully 6x2 - 5x - 6
(Total for Question 8 is 2 marks)
9
Solve the equation 6x2 + 5x - 4 = 0 by factorising.
(Total for Question 9 is 2 marks)
10
Solve 2x2 - 7x - 3 = 0 using the quadratic formula. Give your answers correct to 2 decimal places.
(Total for Question 10 is 2 marks)
11
Find the value(s) of k for which the equation x2 + kx + 16 = 0 has equal roots.
(Total for Question 11 is 2 marks)
12
By completing the square, write x2 + 8x + 5 in the form (x+p)2 + q, where p and q are constants.
(Total for Question 12 is 2 marks)
13
By completing the square, express 2x2 - 12x + 5 in the form a(x+p)2 + q, where a, p and q are constants to be found.
(Total for Question 13 is 2 marks)
14
Solve x2 - 10x + 7 = 0 by completing the square. Give your answer in simplified surd form.
(Total for Question 14 is 3 marks)
15
Prove that x2 - 6x + 12 > 2 for all real values of x.
(Total for Question 15 is 3 marks)
16
Find the set of values of k for which the equation x2 + 4x + k = 0 has two distinct real roots.
(Total for Question 16 is 3 marks)
17
A rectangular photo frame has length (x + 4) cm and width x cm. The area of the frame is 96 cm2. Form an equation in x, and solve it to find the width of the frame.
(Total for Question 17 is 3 marks)
18
The equation x2 - 2kx + (k2 - 9) = 0, where k is a constant, is to be solved for x. Show that this equation always has two distinct real roots for any value of k.
(Total for Question 18 is 4 marks)
19
Solve the equation x4 - 10x2 + 9 = 0.
(Total for Question 19 is 5 marks)
Mark scheme · A3D Quadratic Equations: Factorising, the Formula and Completing the Square: Fluency and Exam Drill
Question 1
B1 A cao
Answer: A
Question 2
B1 (x-8)(x+5) cao
Answer: (x-8)(x+5)
Question 3
B1 (2x-5)(2x+5) cao
Answer: (2x-5)(2x+5)
Question 4
B1 3(x-3)(x+3) cao
Answer: 3(x-3)(x+3)
Question 5
B1 x = -6 and x = 4 (both values) cao
Answer: x = -6 or x = 4
Question 6
B1 x = 5 and x = 6 (both values) cao
Answer: x = 5 or x = 6
Question 7
B1 (3, 7) cao
Answer: Minimum point (3, 7)
Question 8
M1 correct split of the middle term using ac = -36, e.g. 6x2 - 9x + 4x - 6
A1 (3x+2)(2x-3) cao
Answer: (3x+2)(2x-3)
Question 9
M1 attempt to factorise using ac = -24, splitting 5x as 8x - 3x
A1 x = 1/2 and x = -4/3 (both values) cao
Answer: x = 1/2 or x = -4/3
Question 10
M1 correct substitution into x = (-b ± √b2-4ac) / (2a) with a=2, b=-7, c=-3
A1 x = 3.89 and x = -0.39 (both awrt 2dp)
Answer: x = 3.89 or x = -0.39 (2dp)
Question 11
M1 sets discriminant k2 - 4(1)(16) = 0 and rearranges to k2 = 64
A1 k = 8 or k = -8 (both values) oe cao
Answer: k = 8 or k = -8
Question 12
M1 (x+4)2 seen
A1 (x+4)2 - 11 cao
Answer: (x+4)2 - 11
Question 13
M1 takes out factor 2: 2(x2-6x)+5, then 2[(x-3)2-9]+5
A1 2(x-3)2 - 13 cao (a=2, p=-3, q=-13)
Answer: 2(x-3)2 - 13
Question 14
M1 (x-5)2 - 25 + 7 (= 0) oe
M1 (x-5)2 = 18
A1 x = 5 + 3sqrt(2) and x = 5 - 3sqrt(2) fully simplified surd form cao