Question 1
Factorise and thus solve the following quadratic equations, finding both values of x:
1(a) 3x^2+10x-8=0
ANSWER: Multiple choice type 1
A: x = \dfrac{2}{3}, x= -4
B: x = - \dfrac{2}{3}, x= 4
C: x = \dfrac{2}{3}, x= 4
D: x = -\dfrac{2}{3}, x= -4
Answer: A
Workings:
(3x-2)(x+4)=0
x = \dfrac{2}{3} or x= -4
Marks = 3
1(b) 3x^2+39x+126=0
ANSWER: Multiple Answers (Type 2)
Answer: x=-6, x=-7
Workings:
x^2+13x+42=0
(x+6)(x+7)=0[/latex]
x=-6 or x=-7
Marks = 3
1(c) 8x^2+ 46x + 30=0
ANSWER: Multiple choice type 1
A: x= -\dfrac{3}{4}, x=-5
B: x= -\dfrac{3}{4}, x=5
C: x= \dfrac{3}{4}, x=-5
D: x= \dfrac{3}{4}, x=5
Answer: A
Workings:
(8x + 6)(x + 5)=0
x= -\dfrac{3}{4} or x=-5
Marks = 3
1(d) 8x^2+10x+56 = 7x^2+67
ANSWER: Multiple Answers (Type 2)
Answer: x=1 or x=-11
Workings:
x^2+10x-11=0
(x-1)(x+11)=0
x=1 or x=-11
Marks = 3
Question 2
The triangular prism chocolate box shown below has a volume of 140 cm^3.
Determine the only viable length of x.
ANSWER: Simple Text Answer
Answer: x = 2
Workings:
Area \space of \space Triangle = \dfrac{1}{2}base \times height = \dfrac{1}{2}(x+3)(4) = 2x+6
Volume \space of \space Prism = 7x×2x+6 = 14x^2+42x
14x^2+42x=140
x^2+3x-10=0
(x+5)(x-2)=0
x=2 as a length cannot be negative.
Marks = 5
Question 3
Factorise and thus solve the following quadratic equations, finding both values of x:
3(a) 2x^2+ 14x + 24=0
ANSWER: Multiple Answers (Type 2)
x=-3
x=-4
Workings:
2x^2+ 14x + 24 = (2x + 6)(x + 4)
(2x + 6)(x + 4)=0
x = -3 and x = -4
Marks = 2
3(b) 3x^2+ 13x + 14=0
ANSWER: Multiple Choice
Answer:x=-\dfrac{7}{3} and x=-2
x=-\dfrac{3}{7} and x=2
x=-{3} and x=-2
x=-\dfrac{7}{3} and x=2
Workings:
3x^2+ 13x + 14 = (3x + 7)(x + 2)
(3x + 7)(x + 2)=0
x=-\dfrac{7}{3} and x=-2
Marks = 2
3(c) 3x^2+ 30x + 48=0
ANSWER: Multiple Answers (Type 2)
x=-2
x=-8
Workings:
3x^2+ 30x + 48 = (3x+6)(x+8)
(3x+6)(x+8)=0
x = -2 and x=-8
Marks = 2
3(d) 5x^2+ 39x + 28=0
Give your answer as a decimal where appropriate
ANSWER: Multiple Answers (Type 2)
x=-0.8
x=-7
Workings:
5x^2+ 39x + 28 = (5x+4)(x+7)
(5x+4)(x+7)=0
x=-0.8 and x=-7
Marks = 2
3(e) 5x^2+ 27x + 10=0
Give your answer as a decimal where appropriate
ANSWER: Multiple Answers (Type 2)
x= -0.4
x=-5
Workings:
5x^2+ 27x + 10 = (5x+2)(x+5)
(5x+2)(x+5)=0
x= -0.4 and x=-5
Marks = 2
Question 4
Factorise and thus solve the following quadratic equations, finding both values of x:
4(a) 4x^2+ 20x + 16
ANSWER: Multiple Answers (Type 2)
x=-4
x=-1
Workings:
4x^2+ 20x + 16 = (2x+8)(2x+2)
(2x+8)(2x+2)=0
or
4x^2+ 20x + 16 = (4x+4)(x+4)
(4x+4)(x+4)=0
x=-4 and x=-1
Marks = 2
4(b) 6x^2+ 32x + 42
ANSWER: Multiple Choice
Answer: x=-\dfrac{7}{3} and x=-2
x=-\dfrac{7}{3} and x=-3
x=-\dfrac{3}{7} and x=-3
x=-7 and x=-6
Workings:
6x^2+ 32x + 42 = (3x + 7)(2x + 6)
(3x + 7)(2x + 6)=0
x=-\dfrac{7}{3} and x=-3
Marks = 2
4(c) 4x^2 + 18x + 8
Give your answer as a decimal where appropriate
ANSWER: Multiple Answers (Type 2)
x=0.5
x=-4
Workings:
4x^2 + 18x + 8 = (4x + 2)(x + 4)
(4x + 2)(x + 4)=0
x=-0.5 and x=-4
Marks = 2
4(d) 9x^2+ 24x + 16
ANSWER: Multiple choice
Answer: x=-\dfrac{4}{3}
x=-4 and x=-3
x=4 and x=3
x=-\dfrac{3}{4}
Workings:
9x^2+ 24x + 16 = (3x+4)(3x+4)
(3x+4)(3x+4)=0
x=-\dfrac{4}{3}
Marks = 2
Question 5
Factorise and thus solve the following quadratic equations, finding both values of x:
5(a) 2x^2- 18x + 16
ANSWER: Multiple Answers (Type 2)
x=1
x=8
Workings:
2x^2- 18x + 16 = (2x - 2)(x - 8)
(2x - 2)(x - 8)=0
x=1 and x=8
Marks = 2
5(b) 3x^2- 20x + 12
ANSWER: Multiple Answers (Type 2)
Answer: x=\dfrac{2}{3} and x=6
x=\dfrac{3}{2} and x=6
x=\dfrac{3}{2} and x=-6
x=3 and x=6
Workings:
3x^2- 20x + 12 = (3x - 2)(x - 6)
(3x - 2)(x - 6)=0
x=\dfrac{2}{3} and x=6
Marks = 2
5(c) 8x^2- 26x + 6
Give your answer as a decimal where appropriate
ANSWER: Multiple Answers (Type 2)
x=0.25
x=3
Workings:
8x^2- 26x + 6 = (8x - 2)(x-3)
(8x - 2)(x-3)=0
x=0.25 and x=3
Marks = 2
Question 6
Factorise and thus solve the following quadratic equation, finding both values of x:
x^2-64=0
ANSWER: multiple answers
x=8
x=-8
Workings:
x^2-64 = (x+8)(x-8)
(x+8)(x-8)=0
x=8 and x=-8
Marks = 1