U
3
V
Applications
There are many problems that can be solved by equations like those we have just
discussed.
366
Chapter 5
Factoring
5-46
E X A M P L E
7
Area of a garden
Merida’s garden has a rectangular shape with a length that is 1 foot longer than twice the
width. If the area of the garden is 55 square feet, then what are the dimensions of the garden?
Solution
If
x
represents the width of the garden, then 2
x
1 represents the length. See Fig. 5.1.
Because the area of a rectangle is the length times the width, we can write the equation
x
(2
x
1)
55.
We must have zero on the right-hand side of the equation to use the zero factor property.
So we rewrite the equation and then factor:
2
x
2
x
55
0
(2
x
11)(
x
5)
0
Factor.
2
x
11
0
or
x
5
0
Zero factor property
x
1
2
1
or
x
5
The width is certainly not
1
2
1
. So we use
x
5 to get the length:
2
x
1
2(5)
1
11
We check by multiplying 11 feet and 5 feet to get the area of 55 square feet. So the width
is 5 ft, and the length is 11 ft.
Now do Exercises 65–66
x
ft
2
x
1
ft
Figure 5.1
U
Helpful Hint
V
To prove the Pythagorean theorem
start with two identical squares with
sides of length
a
b
, and partition
them as shown.
There are eight identical triangles in
the diagram. Erasing four of them
from each original square will leave
smaller squares with areas
a
2
,
b
2
,
and
c
2
. Since the original squares had
equal areas, the remaining areas
must be equal. So
a
2
b
2
c
2
.
b
a
c
c
c
c
c
2
a
b
b
a
a
b
b
2
b
b
b
c
c
b
a
a
a
a
a
2
Figure 5.2
c
b
a
Hypotenuse
Legs

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5-47
5.6
Solving Quadratic Equations by Factoring
367
The hypotenuse is the longest side of a right triangle. So if the lengths
of the sides of a right triangle are 5 meters, 12 meters, and 13 meters,
then the length of the hypotenuse is 13 meters, and 5
2
12
2
13
2
.
CAUTION
E X A M P L E
8
Using the Pythagorean theorem
The length of a rectangle is 1 meter longer than the width, and the diagonal measures
5 meters. What are the length and width?
Solution
If
x
represents the width of the rectangle, then
x
1 represents the length. Because the two
sides are the legs of a right triangle, we can use the Pythagorean theorem to get a relation-
ship between the length, width, and diagonal. See Fig. 5.3.
x
2
(
x
1)
2
5
2
Pythagorean theorem
x
2
x
2
2
x
1
25
Simplify.
2
x
2
2
x
24
0
x
2
x
12
0
Divide each side by 2.
(
x
3)(
x
4)
0
x
3
0
or
x
4
0
Zero factor property
x
3
or
x
4
The length cannot be negative.
x
1
4
To check this answer, we compute 3
2
4
2
5
2
, or 9
16
25. So the rectangle is
3 meters by 4 meters.
Now do Exercises 67–68
Figure 5.3
5
x
x
1
Warm-Ups
▼
Fill in the blank.
1.
A
equation has the form
ax
2
bx
c
0
where
a
0.
2.
A
equation is two equations connected with
the word “or.”
3.
The
property says that if
ab
0, then
a
0
or
b
0.
4.
Some quadratic equations can be solved by
.
5.
We do not usually
each side of an equation by
a variable.