Question 1:
The following quadratics can be expressed in the form a(x + b)^2 + c
Give the values of a, b and c in each case.
1(a) 2x^2+8x+10
ANSWER: Multiple Answers (Type 1)
Answer: a=2, b=2, c=2
Workings:
2x^2+8x+10 = 2(x^2 +4x)+10
=2((x+2)^2 -4)+10
=2(x+2)^2+2
Marks = 2
1(b) 2x^2-16x-2
ANSWER: Multiple Answers (Type 1)
Answer: a=2, b=-4, c=-34
Workings:
2x^2-16x-2 = 2(x^2-8x)-2
=2((x-4)^2-16)-2
=2(x-4)^2-34
Marks = 2
1(c) 3x^2-24x+6
ANSWER: Multiple Answers (Type 1)
Answer: a=3, b=-8, c=-42
Workings:
3x^2-24x+6 = 3(x^2-8x)+6
3((x-4)^2-16)+6
=3(x-4)^2-42
Marks = 2
Question 2
The following quadratics can be expressed in the form a(x + b)^2 + c
Give the values of a, b and c in each case.
2(a) 3x^2+18x-1
ANSWER: Multiple Answers (Type 1)
Answer: a=3, b=3, c=-28
Workings:
3x^2+18x-1= 3(x^2+6x)-1
=3((x+3)^2-9)-1
=3(x+3)^2-28
Marks = 2
2(b) 4x^2-8x-8
ANSWER: Multiple Answers (Type 1)
Answer: a=4, b=-1, c=-12
Workings:
4x^2-8x-8 = 4(x^2-2x)-8
= 4((x-1)^2-1)-8
=4(x-1)^2-12
Marks =2
2(c) 6x^2-24x-8
ANSWER: Multiple Answers (Type 1)
Answer: a=6, b=-2, c=-12
Workings:
6x^2-24x-8= 6(x^2-4x)-8
=6((x-2)^2-4)-8
=6(x-2)^2-32
Marks = 2
Question 3:
The following quadratics can be expressed in the form a(x + b)^2 + c
Give the values of a, b and c in each case.
3(a) 3x^2+6x -8
ANSWER: Multiple Answers (Type 1)
Answer: a=3, b=1, c=-11
Workings:
3x^2+6x -8 = 3(x^2+2x)-8
=3((x+1)^2-1)-8
=3(x+1)^2-11
Marks = 2
3(b) 5x^2-20x-12
ANSWER: Multiple Answers (Type 1)
Answer: a=5, b=2, c=-32
Workings:
5x^2-20x-12 = 5(x^2--4x)-12
5((x-2)^2-4)-12
5(x-2)^2-32
Marks = 2
3(c) 8x^2+32x-12
ANSWER: Multiple Answers (Type 1)
Answer: a=8, b=2, c=-44
Workings:
8x^2+32x-12 = 8(x^2+4x)-12
8((x+2)^2-4)-12
8(x+2)^2-44
Marks = 2
Question 4:
A rectangle has sides of x and (x−2) cm.
The area of the rectangle is 16 cm^2.
4(a) Create an expression for the area of the rectangle.
Hence form an equation in the form a(x+b)^2+c=0 and give the values of a, b and c
ANSWER: Multiple Answers (Type 1)
Answer: a=2, b=1, c=-18
Workings:
2x(x-2)=16
2x(x-2)-16=0
2x^2-4x-16=0
2(x^2-2x)-16=0
2((x-1)^2-1)-16=0-
2(x-1)^2-18=0
Marks = 3
4(b) Hence, or otherwise, find the perimeter of the rectangle.
ANSWER: Simple text answer
Answer: 20 cm
Workings:
(x-1)^2=9
x=1 \pm 3
x=4
Perimeter, P=8+8+2+2=20 cm
Marks = 3
Question 5:
A small farmers field is shown below.
Given that the area of the field is 27 m^2.
Find the perimeter of the field in meters.
Give your answer in the form a\sqrt{10} + b where a and b are integers.
ANSWER: Multiple choice (Type 1)
A: 9\sqrt{10}+6
B: 9\sqrt{5}-6
C: 18\sqrt{5}+6
D: 18\sqrt{5}-6
Answer: B
Workings:
2x(x+6)=27
2x^2+12x-27=0
2[(x+3)^2-9]-27=0
2(x+3)^2-18-27=0
2(x+3)^2=45
x+3= \pm{\sqrt{\dfrac{45}{2}}}
x=-3 \pm\sqrt{\dfrac{45}{2}}=\dfrac{-6+3\sqrt{10}}{2}
Perimeter, P=2x+2x+(x+6)+(x+6)=6x+12
6x+12=6\times \bigg(\dfrac{-6+3\sqrt{10}}{2}\bigg)+12=-6+9\sqrt{10}
Marks = 6