2. 1. When the square of a certain number is diminished by 9
times the number the result is 36. Find the number.
2
square of a certain number = x
9 times the number = 9 x
2
x 9x 36
2
x 9x 36 0
(x 12 )( x 3) 0
x 12 or 3

3. 1. When the square of a certain number is diminished by 8
times the number the result is 9. Find the number.
2
square of a certain number = x
8 times the number = 8 x
2
x 8x 9
2
x 8x 9 0
(x 9 )( x 1) 0
x 9 or 1

4. 2. A certain number added to its square is 30. Find the
number.
A certain number = x
2
its square = x
2
x x 30
2
x x 30 0
(x 6 )( x 5 ) 0
x ( 6 ) or 5

5. 2. A certain number added to its square is 56. Find the
number.
A certain number = x
2
its square = x
2
x x 56
2
x x 56 0
(x 8 )( x 7 ) 0
x ( 8 ) or 7

6. 3. The square of a number exceeds the number by 72. Find
the number.
a number = x
2
square of a number = x
2
x x 72
2
x x 72 0
( x 9 )( x 8 ) 0
x 9 or 8

7. 4. Find two positive numbers whose ratio is 2:3 and whose
product is 600.
First numbers = x
numbers whose ratio is 2:3 = 2x:3x
3
x: x
2
3
product is 600 = x ( x) 600
2
2
3x 2
600 x 400
2 2
x 400 0
2
3x 1200
2 2
x 20 0
2 1200
x
3

8. (x 20 )( x 20 ) 0
x 20 or 20
positive numbers x 20
3 3
two positive numbers x : x 20 : ( 20 ) 20 : 30
2 2

9. 5. The product of two consecutive odd integers is 99. Find
the integers.
odd integers = x
consecutive odd integers = x 2
x( x 2) 99 If x = 9
2 consecutive odd integers
x 2 x 99 = 9+2=11
2
x 2 x 99 0 If x = -11
(x 9 )( x 11 ) 0 consecutive odd integers
= -11+2=-9
x 9 or 11

10. 6. Find two consecutive positive integers such that the square
of the first is decreased by 17 equals 4 times the second.
First number = x
Second number = x 1
2
x 17 4 ( x 1)
2
x 17 4 ( x 1) 0
2
x 17 4 x 4 0
2
x 4 x 21 0 First number = 7
( x 7 )( x 3) 0 Second number = 7+1=8
x 7 or 3
x 7

11. 7. The ages of three family children can be expressed as
consecutive integers. The square of the age of the youngest
child is 4 more than eight times the age of the oldest child.
Find the ages of the three children.
youngest child = x
oldest child = x 2
2
x 4 8( x 2 )
2
x 4 8 x 16
2
x 8 x 20 0
( x 2 )( x 10 ) 0
x 2or 10
x 10
The ages of three family children = 10,11,12 year old

12. 8. Find three consecutive odd integers such that the square of
the first increased the product of the other two is 224.
2
x (x 2 )( x 4) 224
2 2
x x 6 x 8 224
2
2x 6 x 216 0
2
x 3 x 108 0
(x 12 )( x 9 ) 0
x 12 or 9
x 9
three consecutive odd integers = 9,10,11

13. 9. The sum of the squares of two consecutive integers is 113.
Find the integers.
2 2
x ( x 1) 113
2 2
x x 2 x 1 113
2
2x 2 x 112 0
2
x x 56 0
( x 7 )( x 8 ) 0
x 7 or 8
If x= 7 two consecutive integers is 7 and 8
If x= -8 two consecutive integers is -8 and -7

14. 10. The sum of the squares of three consecutive integers is
302. Find the integers.
2 2 2
x (x 1) ( x 2) 302
2 2 2
x (x 2 x 1) ( x 4 x 4) 302
2
3x 6 x 297 0
2
x 2 x 99 0
( x 11 )( x 9 ) 0
x 11 or 9
If x= -11 three consecutive integers is -11,-10,-9
If x= 9 three consecutive integers is 9,10,11

15. 11. Two integers differ by 8 and the sum of their squares is
130.
2 2
x (x 8) 130
2 2
x x 16 x 64 130
2
2x 16 x 66 0
2
x 8 x 33 0
( x 3)( x 11 ) 0
x 3 or 11
If integers is -3 two integers is -3 and -11
If integers is 11 two integers is 11 and 3

16. 12. rectangle is 4 m longer than it is wide, its area is 192 m2.
Find the dimensions of the rectangle.
x ( x 4 ) 192
2
x 4 x 192
2
x 4 x 192 0
( x 16 )( x 12 ) 0
x 16 or 12
x 12
12 Dimensions of the rectangle
= 12 x 16 = 192
12 4

17. 13. right triangle has a hypotenuse of 10 m. If one of the sides is 2 m
longer than the other, find the dimensions of the triangle.
2 2 2
c a b
2 2 2
10 x (x 2)
2 2 2
10 x x 4x 4
2 2
0 x x 4x 4 100
2
0 2x 4 x 96
0 (x 6 )( x 8 )
x 6 or 8
x 6
the dimensions is 6 and 8

18. 14. The sum of the squares of three consecutive even integers
is 980. Determine the integers
2 2 2
980 x (x 2) (x 4)
2 2 2
980 x (x 4x 4) (x 8x 16 )
2
980 3x 12 x 20
2
0 3x 12 x 20 980
2
0 x 4x 320
0 (x 16 )( x 20 )
x 16 or 20
If x = 16 three consecutive even integers is 16,18,20
If x = -20 three consecutive even integers is -20,-18,-16

19. 15. Two integers differ by 4. Find the integers if the sum of
their squares is 250. same No.11)
2 2
x (x 4) 250
2 2
x x 8 x 16 250
2
2x 8 x 234 0
2
x 4 x 117 0
( x 13 )( x 9 ) 0
x 13 or 9
If integers is 13 two integers is 13 and 9
If integers is -9 two integers is -9 and -13

20. 16. Find three consecutive integers such that the product of
the first and the last is one less than five times the middle
integer.
x ( x 2) 1 5 ( x 1)
2
x 2x 1 5x 5
2
x 2x 1 5x 5 0
2
x 3x 4 0
( x 1)( x 4) 0
x 1 or 4
If x = -1 three consecutive integers = -1,0,1
If x = 4 three consecutive integers = 4,5,6

21. 17. A rectangle is three times as long as it is wide. Its area has
measure equal to ten times its width measure plus twenty-
five. What is the width?
x ( 3 x ) 10 x 25
2
3x 10 x 25 0
( 3 x 5 )( x 5 ) 0
5
x or 5
3
x 5

22. 18. Richie walked 15 m diagonally across a rectangular field.
He then returned to his starting position along the outside of
the field. The total distance he walked was 36 m. What are
the dimensions of the field?
2 2 2
15 ( 21 x) x
2 2
225 441 42 x x x
2
0 2x 42 x 216
2
0 x 21 x 108
0 (x 9 )( x 12 )
x 9or 12
The dimensions of the field= 9x12=108

23. 19. A square picture with sides of length 20 cm is to be
mounted centrally upon a square frame. If the area of the
border of the frame equals the area of the picture, find the
width of the border of the frame to the nearest millimetre.
2
20 4 x ( 20 x)
400 2
20 x x
4
2
100 20 x x
2
x 20 x 100 0
2
b b 4 ac
x
2a

24. x 10 5 8 or 10 5 8
x 4 . 142 or 24 . 142
x 4 . 142

25. 20. A square lawn is surrounded by a walk 1 m wide. The
area of the lawn is equal to the area of the walk. Find the
length of the side of the lawn to the nearest tenth of a metre.
2
x 4( x 1)
2
x 4x 4
2
x 4x 4 0
2
b b 4 ac
x
2a
2
( 4) ( 4) 4 1 ( 4)
2 1

26. x 2 2 2 or 2 2 2
x 4 . 828 or 0 . 828
x 4 . 828
x 5

27. 21. A picture 20 cm wide and 10 cm high is to be centrally
mounted on a rectangular frame with total area three times
the area of the picture. Assuming equal margins for all four
sides, find the width of the margin.
2 x 20 x 2 x 10 x 3 20 10
2
4x 60 x 600 0
2
x 15 x 150 0
15 33 15 33
x or
2 2
x 6 . 861 or 2 . 861

28. x 6 . 861 7
The width of the margin= 10+7 = 17

29. 22. A fast food restaurant determines that each 10¢ increase
in the price of a hamburger results in 25 fewer hamburgers
sold. The usual price for a hamburger is $2.00 and the
restaurant sells an average of 300 hamburgers each day.
What price will produce revenue of $637.50?
100¢ = $1
10¢ = $ 0.1
( 300 x)
2 ( 300 x) 0 .1 637 . 5
25
0 .1
( 300 x )( 2 ) 637 . 5
25

30. ( 300 x )(1 . 996 ) 637 . 5
637 . 5
( 300 x)
1 . 996
300 x 319 . 38
x 319 . 38 300
x 19 . 38
x 20

31. 23. George operates his own store, George’s Fashion. A
popular style of pants sells for $50. At that price, George sells
about 20 pairs of pants a week. Experience has taught
George that for every $1 increase in price, he will sell one less
pair of pants per week. In order to break even, George needs
to produce $650/week. How many pairs of pants does
George need to sell to break even?
50 ( 20 x) 650
1000 650 50 x
350
x 7
50
George need to sell to break even =20+7 = 27

32. 24.A biologist predicts that the deer population, P, in a certain national park
can be modelled by 2
P 8x 112 x 5 7 0,
where x is the number of years since 1999.
a) According to the model, how many deer were in the park in 1999.
b) Will the deer population ever reach zero?
c) In what year will the population reach 1000?
a) According to the model, how many deer were in the
park in 1999 x 1
2
p 8 (1) 112 (1) 570
p 466

33. b) Will the deer population ever reach zero?
p 0
2
0 8x 112 x 570

34. c) In what year will the population reach 1000?
2
1000 8x 112 x 570
2
0 8x 112 x 430
2
b b 4 ac 14 411
x x
2a 2
x 17 . 137 or 3 . 137 x 17 . 137
x 18
18 year will the population reach 1000

35. 25. A t-ball player hits a baseball from a tee. The flight of the ball can be modelled by
2
h 4.9t 9t 0.6
where h is the height in metres, and t is the time in seconds.
a) How high is the tee?
b) How long does it take the ball to land?
c) When does the ball reach a height of 4.5 m?
a) How high is the tee? t 0
2
h 4 .9 t 9t 0 .6
2
h 4 .9 ( 0 ) 9(0) 0 .6
h 0 .6

36. b) How long does it take the ball to land?
the ball take to land h 0
2
h 4 .9 t 9t 0 .6
2
0 4 .9 t 9 t 0 .6
2
0 4 .9t 9 t 0 .6
2
0 49 t 90 t 6
2
b b 4 ac
t
2a
2
( 90 ) ( 90 ) 4 ( 49 )( 6 )
t
2 ( 49 )

37. 90 8100 1176
t
2 ( 49 )
45 231
t
( 49 )
t 1 . 9 or 0 . 064
t 1 .9

38. c) When does the ball reach a height of 4.5 m?
h 4 .5
2
4 .5 4 .9 t 9t 0 .6
2
0 4 .9 t 9t 0 .6 4 .5
2
0 4 .9 t 9t 3 .9
2
0 4 .9t 9 t 3 .9
2
0 49 t 90 t 39
2
b b 4 ac
t
2a

39. 2
b b 4 ac
t
2a
45 114
t
49
t 1 . 136 or 0 . 7