Question 1
The following quadratics can be written in the form (x+a)^2+b where a and b are integers.
Give the values of a and b.
1(a) x^2+4x+5
ANSWER: Multiple Answers (Type 1)
Answer: a = 2, b = 1
Workings:
x^2+4x+5 = (x + 2)^2 + b
= (x^2 + 4x + 4) + b
so b = 5 - 4 = 1
(x + 2)^2 + 1
Marks = 2
1(b) x^2-14x-1
ANSWER: Multiple Answers (Type 1)
Answer: a = -7, b = -50
Workings:
x^2-14x-1 = (x - 7)^2 + b
= (x^2 -14x + 49) + b
so b = -1 - 49 = -50
(x - 7)^2 + -50
Marks = 2
1(c) x^2-24x+5
ANSWER: Multiple Answers (Type 1)
Answer: a = -12, b = -139
Workings:
x^2-24x+5 = (x - 12)^2 + b
= (x^2 -24x + 144) + b
so b = 5 - 144 = -139
(x - 12)^2 -139
Marks = 2
Question 2
The following quadratics can be written in the form (x+a)^2+b where a and b are integers.
Give the values of a and b.
2(a) x^2+10x+8
ANSWER: Multiple Answers (Type 1)
Answer: a = 5, b = -17
Workings:
x^2+10x+8 = (x+5)^2 + b
= (x^2+10x+25) + b
so b = 8 - 25 = -17
(x+5)^2 -17
Marks = 2
2(b) x^2-4x+16
ANSWER: Multiple Answers (Type 1)
Answer: a = -2, b = 12
Workings:
x^2-4x+16 = (x-2)^2 + b
= (x^2-4x+4) + b
so b = 16-4 = 12
(x-2)^2 + 12
Marks = 2
2(c) x^2-8x+14
ANSWER: Multiple Answers (Type 1)
Answer: a = -4, b = -2
Workings:
x^2-8x+14 = (x-4)^2 + b
= (x^2-8x+16) + b
so b = 14-16 = -2
(x-4)^2 - 2
Marks = 2
Question 3
The following quadratics can be written in the form (x+a)^2+b where a and b are integers.
Give the values of a and b.
3(a) x^2+6x+20
ANSWER: Multiple Answers (Type 1)
Answer: a = 3, b = 11
Workings:
x^2+6x+20 = (x+3)^2 + b
= (x^2+6x+9) + b
so b = 20-9 = 11
(x+3)^2 + 11
Marks = 2
3(b) x^2+12x-8
ANSWER: Multiple Answers (Type 1)
Answer: a = 6, b = -44
Workings:
x^2+12x-8 = (x+6)^2 + b
= (x^2+12x+36) + b
so b = -8 - 36 = -44
(x+6)^2 -44Marks = 2
3(c) x^2-2x-6
ANSWER: Multiple Answers (Type 1)
Answer: a = -1, b = -7
Workings:
x^2-2x-6 = (x-1)^2 + b
= (x^2-2x+1) + b
so b = -6 - 1 = -7
(x-1)^2 - 7
Marks = 2
Question 4
A rectangle has sides of x and (x−2) cm.
The area of the rectangle is 3 cm^2.
4(a) Create an expression in the form (x + a)^2 + b = 0 and give the values of a and b.
ANSWER: Multiple Answers (Type 1)
Answer: a= -1, b= -4
Workings:
x(x-2)=3
x^2-2x=3
x^2-2x -3=0
(x-1)^2-1-3=0
(x-1)^2-4=0
Marks = 3
4(b) Hence, or otherwise, find the perimeter of the rectangle.
ANSWER: Simple Text Answer
Answer: \text{Perimeter} = 8cm
Workings:
(x-1)^2 = 4
x = 1 \pm\sqrt{4}
x = 3, since length is positive.
Hence, \text{Perimeter} = 2(x-2) + 2(x) = 2(3-2) + 2(3) = 2 + 6 = 8
Marks = 3
Question 5
Bob adds a number, that is larger than 1, to its reciprocal.
His answer is 4.
Find Bob’s number in the form a\pm\sqrt{b}
ANSWER: Multiple Choice (Type 1)
A: x = -2 \pm\sqrt{3}
B: x = 2 \pm\sqrt{3}
C: x = 3 \pm\sqrt{2}
D: x = -3 \pm\sqrt{2}
Answer: B
Workings:
x+\dfrac{1}{x} = 4
x^2 - 4x + 1=0
(x-2)^2 - 3 = 0
(x-2)^2=3
x -2 = \pm\sqrt{3}
x = 2 \pm\sqrt{3}
x = 2 + \sqrt{3}
The only valid solution in this case is x = 2 + \sqrt{3} since the question states that Bob’s number is greater than \textbf{1}
Marks = 4
Question 6
A small farmers field is shown below.
The area of the field is 36 m^2.
Find the perimeter of the field in meters.
Give your answer in the form a\sqrt{5} + b where a and b are integers.
ANSWER: Multiple Choice (Type 1)
A: 6 \sqrt{5} + 24
B: 6 \sqrt{5} + 36
C: 12 \sqrt{5} + 24
D: 12 \sqrt{5} + 24
Answer: C
Workings:
x(x+6)=36
x^2 + 6x -36=0
[(x+3)^2-9]-36=0
(x+3)^2-45=0
(x+3)^2=45
x+3= \pm\sqrt{45}
x=3\pm\sqrt{45} thus x=3+3\sqrt{5}
Perimeter, P: P=x+x+(x+6)+(x+6)=4x+12
4x+12=4\times (3+3\sqrt{5})+12=12\sqrt{5}+24
Marks = 4
THIS IS WRONG!!!!!!