Name: ________________________
Quadratic Applications
General Word Problems/Basic Translating
1
The square of a number decreased by 8 times the
number equals 9. Determine the two possible
values of the number.
2
If the square of a positive number is increased by
five times the number, the result is 36. Find the
number.
3
When a textbook sells for $20, 70 copies are
purchased. Every time the price is increased by
$3, the number of people purchasing the book
decreases by 8. Which equation can be used to
determine the number of $3 increases needed to
make the total revenue from the books equal to
$1426?
(1) (20 − 3𝑥)(70 − 8𝑥) = 1426
(2) (20 − 3𝑥)(70 + 8𝑥) = 1426
(3) (20 + 3𝑥)(70 + 8𝑥) = 1426
(4) (20 + 3𝑥)(70 − 8𝑥) = 1426
Date: ___________
Algebra Common Core
Topic 15
4
A man is 35 years older than his son. The product
of their ages is 884. Determine the age of the man.
[Only an algebraic solution will be accepted.]
5
Tamara has two sisters. One of the sisters is 7
years older than Tamara. The other sister is 3
years younger than Tamara. The product of
Tamara’s sisters’ ages is 24. How old is Tamara?
Consecutive Integers
6
Marsha and Billy have ages that are consecutive
odd integers. The product of their ages is 99. If
Marsha is the youngest, find both of their ages.
7
Find two consecutive negative even integers
whose product is 80.
8
Sam, Jimmy, and Joe are each one year apart. Joe
is the youngest and Jimmy is the oldest. If the
product of Joe’s and Jimmy’s age is 15, find the
ages of all three boys. [Only an algebraic solution
will be accepted.]
9
Write an equation such that the product of the first
and second consecutive integer is equal to four
times the third consecutive integer.
10
Find three consecutive positive integers such that
the square of the third is 51 more than the sum of
the first and second.
11
12
Find three consecutive positive integers such that
the product of the first and second is two more
than three times the third.
13
The sum of the squares of two consecutive
positive odd integers is 34. Find the integers.
14
The sum of the squares of two consecutive
positive integers is 41. Find the integers.
15
The length of the shortest side of a right triangle is
10 inches. The lengths of the other two sides are
represented by consecutive integers. Write an
equation that could be used to find the lengths of
the other sides of the triangle.
Find three consecutive positive integers such that
the square of the first increased by twice the
second is 3 less than four times the third.
Basic Area of Rectangles
16
The length of a rectangular room is 5 less than
double the width, w, of the room. Write an
expression to represent the area of the room.
17
The total area of Jon's backyard is 144 square feet.
He knows that the length is 10 feet more than the
width. Determine algebraically the dimensions of
Jon's backyard.
18
The length of a rectangle is 4 centimeters more
than 3 times its width. If the area of the rectangle
is 15 square centimeters, find the width.
19
A student is painting an accent wall in his room
where the length of the room is 3 ft. more than the
width. The wall has an area of 130 square feet.
What are the length and width, in feet, of the
room?
20
The area of a rectangle is 40. If the length is 6
more than the width, find the dimensions of the
rectangle algebraically.
21
The length of a rectangle is 4 feet more than its
width. If the area of the rectangle is 140 square
feet, find the dimensions of the rectangle.
23
The length of a photograph is 3 inches more than
twice its width. If the area of the photograph is 27
square inches, what are the dimensions of the
photograph?
22
The length of a rectangle is 2 cm less than its
width. If the area of the rectangle is 35 cm2, find
the dimensions of the rectangle.
24
A landscaper is creating a rectangular flower bed
such that the width is half of the length. The area
of the flower bed is 40 square feet. Write and
solve an equation to determine the width of the
flower bed to the nearest tenth of a foot.
25
A rectangular field is 20 yards longer than it is
wide. Its area is 2400 square yards. Find the
dimensions of the field.
Transforming Squares to Rectangles
26
The length of each side of a square is represented
by x meters. A rectangle is formed by increasing
the width of the square by 2 meters and decreasing
the length of the square by 2 meters. The area of
the rectangle formed is 32 square meters. Find the
measure of one side of the original square.
27
The length of a rectangle is 5 inches more than
twice the length of a square. The width is 4 inches
less than the width of the square. If the area of the
rectangle is 15 square inches, find the dimensions
of the square.
Increasing Areas – Comparing original to new
28
A garden is 3 feet by 5 feet. How much will each
side have to be increased in order to increase the
area by 20 square feet?
29 A parking area is 8 meters by 10 meters. How
much will each side have to be increased in
order to increase the area by 88 square meters?
31
New Clarendon Park is undergoing renovations to
its garden. One garden that was originally a square
is being adjusted so that one side is doubled in
length, while the other side is decreased by three
meters. The new rectangular garden will have an
area that is 25% more than the original square
garden.
Write an equation that could be used to determine
the length of a side of the original square garden.
30
Steven has a rectangular patio that measures 9 feet
by 12 feet. He wants to increase the area by 25%
and plans to increase each dimension by equal
lengths, x. Write an equation that can be used to
determine x. [Write as a quadratic equation in
standard form.]
Explain how your equation models the situation.
Determine the area, in square meters, of the new
rectangular garden.
Comparing Squares
32
The side of one square is 2 inches longer than
the side of a second square. If the sum of their
areas is 130 square inches, find the length of the
side of each square.
33 The side of one square is 2 centimeters longer
than the side of a second square. If the sum of
their areas is 100 square centimeters, find the
length of the side of each square.
34 Two floors, each square in form and one 7 feet
wider than the other, contain together 1429
square feet. How many square feet in each?
Perimeter and Area Problems
35
The perimeter of a rectangle is 20 meters and the
area of the rectangle is 16 square meters. Find the
dimensions of the rectangle.
36
The perimeter of a rectangle is 40 feet and the area
is 96 square feet. Determine the dimensions of the
rectangle.
37
Samantha is going to fence a rectangular pasture
for her horse. The amount of fencing to be used is
32 feet. The fenced pasture will have an area of 63
square feet. Determine the length and width of the
pasture.
38
A contractor has 48 meters of fencing that he is
going to use as the perimeter of a rectangular
garden. The length of one side of the garden is
represented by x, and the area of the garden is 128
square meters. Determine, algebraically, the
dimensions of the garden in meters.
39
Carlos is going to fence a rectangular space for his
pit bulls. The amount of fencing to be used is 44
feet. The fenced space will have an area of 120
square feet. Determine, algebraically, the
dimensions of the space for the pit bulls, in feet..
Border Problems
40
A rectangular picture measuring 8 inches by 10
inches is put into a picture frame with a border of
constant width. If the area of the frame, including
the picture, is 144 square inches, write an equation
for the width of the border. [Write the quadratic
equation in standard form.]
41
A printer wants to put 48 square inches of text into
a rectangle of a 9 inch by 12 inch sheet of paper.
She wants the text to be surrounded by a border of
constant width. Write an equation that could be
used to find the width of the border (x).
42
A 4 inch by 6 inch photo is put into a picture
frame with a border of constant width. If the area
of the frame, including the picture, is 80 square
inches, find an equation and solve for the width of
the border.
43
The Smiths’ have decided to put a paved walkway
of uniform width around their swimming pool.
The pool is a rectangular pool that measures 12
feet by 20 feet. The area of the walkway will be
68 square feet. Find the width of the walkway.
44
A garden measuring 12 meters by 16 meters is to
have a pedestrian pathway that is x meters wide
installed all the way around it, increasing the total
area to 285 square meters.
45
A rectangular picture measures 8 inches by 10
inches. Simon wants to build a wooden frame for
the picture so that the framed picture takes up an
area of 224 square inches on the wall. The pieces
of wood that he uses to build the frame all have
the same width.
Write an equation that can be used to determine the
width, x, of the pathway.
Write an equation that can be used to determine
the width of the pieces of wood for the frame
Simon could create.
Explain how this equation models the situation.
Explain how this equation models the situation.
Use this equation to solve for the width of the
pathway.
Solve the equation to determine the width of the
frame.