Download QUADRATIC EQUATIONS WORD PROBLEMS Sheet 3

QUADRATIC EQUATIONS
WORD PROBLEMS
Sheet 3 - Solutions
1-
The sum of a number and its square is 72. Find the number.
*
Let
x
be a number
x2 + x = 72
x2 + x - 72 = 0
(x + 9)(x - 8) = 0
x+9=0
x-8=0
x = -9
x=8
The number is -9 or 8.
2-
If you square 5 more than a positive number, the result is 88 more than
12 times the number. Find the number.
*
Let
x
+ve number
(x + 5)2 = 12x + 88
x2 + 10x + 25 = 12x + 88
x2 - 2x - 63 = 0
(x - 9)(x + 7) = 0
x-9=0
x+7=0
x=9
x = -7
The number is 9.
3-
The sum of two numbers is 12 and the sum of their squares is 80.
Find the numbers.
*
Let
x
1st number
12 - x 2nd number
x2 + (12 - x)2 = 80
x2 + 144 - 24x + x2 = 80
2x2 - 24x + 64 = 0
x2 - 12x + 32 = 0
(x - 8)(x - 4) = 0
x-8=0
x-4=0
x=8
x=4
The numbers are 4, 8.
Quadratic Equations - Word Problems - Sheet 3 - Solutions
page 1
4-
The length of a rectangle is 4 cm more than its width.
The area of the rectangle is 96 sq. cm. Find its dimensions.
*
Let
x
x+4
width
length
x(x + 4) = 96
x2 + 4x = 96
x2 + 4x - 96 = 0
(x + 12)(x - 8) = 0
x = -12
x=8
The dimensions are: width 8 cm, length 12 cm.
5-
The perimeter of a rectangle is 30 cm. Its area is 54 cm2.
Find the dimensions of this rectangle.
*
Let
x
width
15 - x length
x(15 - x) = 54
15x - x2 - 54 = 0
x2 - 15x + 54 = 0
(x - 6)(x - 9) = 0
x=6
x=9
The dimensions are: 6 cm by 9 cm
6-
One number is 3 more than another. The sum of their squares
is 149. Find the numbers.
*
Let
x
1st number
x + 3 2nd number
x2 + (x + 3)2 = 149
x2 + x2 + 6x + 9 = 149
2x2 + 6x - 140 = 0
x2 + 3x - 70 = 0
(x + 10)(x - 7) = 0
x + 10 = 0
x-7=0
x = -10
x=7
The numbers are: -10, -7 or 7, 10
7-
The product of two consecutive positive numbers is 132.
Find the numbers.
*
Let
x
+ve first number
x + 1 +ve number
x(x + 1) = 132
x2 + x - 132 = 0
(x + 12)(x - 11) = 0
x + 12 = 0
x - 11 = 0
x = -12
x = 11
Quadratic Equations - Word Problems - Sheet 3 - Solutions
page 2
The numbers are 11, 12.
Quadratic Equations - Word Problems - Sheet 3 - Solutions
page 3
8-
The sum of the squares of two consecutive positive even integers
is 340. Find the integers.
*
Let
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 = 0
x - 12 = 0
x =-14
x = 12
x
+ve number
x + 2 +ve number
The numbers are 12, 14
9-
The length of a rectangle is 1 m longer than 3 times its width.
The area is 80 m2. Find the dimensions of this rectangle.
*
Let
x
3x + 1
width
length
x(3x + 1) = 80
3x2 + x - 80 = 0
(3x + 16)(x - 5) = 0
x = -16/3
x=5
The dimensions are: width 5 m, length 16 m.
10-
The length of a rectangle is 3 times its width. If the width is increased
by 2 cm and the length is increased by 4 cm, its area will be doubled.
Find the original dimensions of the rectangle.
*
Let
x
original width
3x
original length
x + 2 new width
3x + 4 new length
(x + 2)(3x + 4) = 2(3x2)
3x2 + 10x + 8 = 6x2
3x2 - 10x - 8 = 0
(3x + 2)(x - 4) = 0
x = -2/3
x=4
The dimensions are 4 cm by 12 cm.
Quadratic Equations - Word Problems - Sheet 3 - Solutions
page 4
11-
The sum of a certain positive number and three times its square
is 52. Find the number.
*
Let
x
+ve number
3x2 + x = 52
3x2 + x - 52 = 0
(3x + 13(x - 4) = 0
3x + 13 = 0 x - 4 = 0
x = -13/3
x-4
The number is 4.
12-
If one side of a square is increased by 10 cm and another side is
increased by 5 cm, a rectangle is formed with an area 3 times the area
of the square. Find the length of the side of the square.
*
Let
x
Side of square
x + 10 width of rectangle
x + 5 length of rectangle
(x + 5)(x + 10) = 3x2
x2 + 15x + 50 = 3x2
2x2 - 15x - 50 = 0
(2x + 5)(x - 10) = 0
x = -5/2
x = 10
The square had a side of length 10 m.
13-
If 12 is subtracted from 3 times the square of a positive number, the
result is 5 times the number. Find the number.
*
Let
x
+ve number
3x2 - 12 = 5x
3x2 - 5x - 12 = 0
(3x + 4)(x - 3) = 0
3x + 4 = 0
x-3=0
x = -4/3
x=3
The number is 3.
14-
Given two consecutive integers, if 3 times the first is increased by the
square of the second, the result is 85. Find these numbers.
*
Let
x
1st number
x + 1 2nd number
3x + (x + 1)2 = 85
x2 + 2x + 1 + 3x = 85
x2 + 5x - 84 = 0
(x + 12)(x - 7) = 0
x + 12 = 0
x-7=0
x = -12
x=7
The numbers are 7, 8 or -12, -11.
Quadratic Equations - Word Problems - Sheet 3 - Solutions
page 5
15-
The perimeter of a rectangle is 60 m, its area is 200 m2.
What are the dimensions of the rectangle?
*
Let
x
width
30 - x length
x(30 - x) = 200
30x - x2 - 200 = 0
x2 - 30x + 200 = 0
(x - 10)(x - 20) = 0
x = 10
x = 20
The dimensions are 10 m by 20 m.
16-
A negative number and its square add up to 6. Find the number.
*
Let
x
-ve number
x2 + x = 6
x2 + x - 6 = 0
(x + 3)(x - 2) = 0
x+3=0
x-2=0
x = -3
x=2
The number is -3.
Quadratic Equations - Word Problems - Sheet 3 - Solutions
page 6