Download QUADRATIC EQUATIONS WORD PROBLEMS Sheet 2

QUADRATIC EQUATIONS
WORD PROBLEMS
Sheet 2 - Solutions
1-
Find two consecutive positive integers whose product is 72.
*
Let
x
+ve number
x + 1 + ve number
The numbers are 8, 9.
x(x + 1) = 72
x2 + x = 72
x2 + x - 72 = 0
(x + 9)(x - 8) = 0
x = -9
x=8
2-
Find two consecutive negative integers whose product is 156.
*
Let
x
-ve number
x + 1 -ve number
x(x + 1) = 156
x2 + x = 156
x2 + x - 156 = 0
(x + 13)(x - 12) = 0
x = -13
x = 12
The numbers are -13, -12
3-
The sum of the squares of two consecutive negative odd integers
is 202. Find the integers.
*
Let
x
-ve number
x + 1 -ve number
x2 + (x + 1)2 = 202
x2 + x2 + 2x + 1 = 202
2x2 + 2x - 201 = 0
x2 + 2x - 99 = 0
(x + 11)(x -9) = 0
x = -11
x=9
The numbers are: -11, -9
4-
The sum of the squares of two consecutive positive even integers
is 340. Find the integers.
*
Let
x
+ve number
x + 2 +ve number
x2 + (x + 2)2 = 340
x2 + x2 + 4x + 4 = 340
2x2 + 4x - 336 =0
x2 + 2x - 168 = 0
(x + 14)(x - 12) = 0
x =-14
x = 12
The numbers are 12, 14
Quadratic Equations - Word Problems - Sheet 2 - Solutions
page 1
5-
Find the dimensions of a rectangle with area 10 cm2 if its width is
3 cm less than its length.
*
Let
x
x-3
width
length
x(x - 3) = 10
x2 - 3x = 10
x2 - 3x - 10 = 0
(x - 5)(x + 2) = 0
x=5
x = -2
The dimensions are: width 5 cm, length 2 cm.
6-
The length of a rectangle is 5 meters longer than twice its width.
If its area is 33 sq. m. find its dimensions.
*
Let
x
width
2x + 5 be length
x(2x + 5) = 33
2x2 + 5x - 33 = 0
(2x + 11)(x - 3) = 0
x = -11/2
x=3
The dimensions are: width 3 m, length 11 m
7-
The perimeter of a rectangular wildlife preserve is 8 km, and its area
is 3 km2. Find the dimensions of the preserve.
*
Let
x
4-x
width
length
x(4 - x) = 3
4x - x2 - 3 = 0
x2 - 4x + 3 = 0
(x - 3)(x - 1) = 0
x=3
x=1
The dimensions are: 3 km by 1 km.
8-
When the dimensions of a 2cm by 5 cm rectangle were increased by
equal amounts, its area increased by 18 cm2. Find the dimensions of
the new rectangle.
*
Let
x
the increment
x + 2 new dimension
x + 5 new dimension
(x + 2)(x + 5) = 10 + 18
x2 + 7x + 10 = 28
x2 + 7x - 18 = 0
(x + 9)(x - 2) = 0
x = -9
x=2
The dimensions of the new rectangle are: 4 cm and 7 cm
Quadratic Equations - Word Problems - Sheet 2 - Solutions
page 2
9-
The sum of the squares of three consecutive positive integers is equal
to the sum of the squares of the next two integers. What are the five
integers?
*
Let
x
x+1
x+2
x+3
x+4
+ve first number
+ve second number
+ve third number
+ve fourth number
+ve fifth number
x2 + (x + 1)2 + (x + 2)2 = (x + 3)2 +(x + 4)2
2
2
x + x + 2x + 1 + x2 + 4x + 4 = x2 + 6x + 9 + x2 + 8x + 16
3x2 + 6x + 5 = 2x2 + 14x + 25
x2 - 8x - 20 = 0
(x - 10)(x + 2) = 0
x = 10
x = -2
The numbers are: 10, 11, 12, 13, 14
10-
A 6 m by 9 m rug covers half of the floor area of a room and leaves a
uniform strip of bare floor around the edges. What are the dimensions
of the room?
*
Let
11-
The page of a book is twice as high as it is wide.
There are margins 2 cm wide at the top and sides and 3 cm wide at the
bottom. Find the dimensions of the page if the area of the printed part is
152 cm2.
*
Let
x
Uniform strip around carpet
2x + 6 dimension of room
(2x + 6)(2x + 9) = 2(6)(9)
2x + 9 dimension of room
4x2 + 30x + 54 = 108
4x2 + 30x - 54 = 0
2x2 + 15x - 27 = 0
(2x - 3)(x + 9) = 0
x = 3/2
x = -9
The dimensions of the room are: 12 m by 9 m.
x
2x
x-4
2x - 5
width of page
height of page
width of print
height of print
(x - 4)(2x - 5) = 152
2x2 - 13x + 20 = 152
2x2 - 13x - 132 = 0
(2x + 11)(x - 12) = 0
x = -11/2
x = 12
The dimensions of the page are: 12 cm by 24 cm.
Quadratic Equations - Word Problems - Sheet 2 - Solutions
page 3
12-
A rectangle is 5 cm longer than it is wide.
The area of the rectangle is 66 cm2. Find its width.
*
Let
x
width
x + 5 be length
x(x + 5) = 66
x2 + 5x - 66 = 0
(x + 11)(x - 6) = 0
x = -11
x=6
The dimensions are: width 6 cm, length 11 cm
13-
A rectangle is 2 m longer than it is wide.
The area of the rectangle is 99 m2. Find its length.
*
Let
x
width
x + 2 length
x(x + 2) = 99
x2 + 2x - 99 = 0
(x + 11)(x - 9) = 0
x = -11
x=9
The dimensions are: width 9 m, length 11 m.
Quadratic Equations - Word Problems - Sheet 2 - Solutions
page 4