When we are given the area of a rectangle, we generally need to use the formula for the area of a rectangle. In this case we need to write expressions for the length, ℓ, and width, w.
Verbal Expression
|
Algebraic Expression
|
the length is 10 ft more than the width
|
ℓ=10+w
|
Now that we have an algebraic expression, we can substitute. Then, let's see where we can go from there.
A=ℓ⋅w
144=(10+w)w
144=10w+w2
0=10w+w2−144
0=w2+10w−144
w2+10w−144=0
Now, we have a quadratic equation that is equal to zero. Let's look for two factors of
-144 that have a sum of
10. Since
ac is negative, the sign of the factors will be different. Furthermore, since
b is positive, the factor with the larger magnitude will be positive too.
Factors
|
Sum of factors
|
-1,144
|
143
|
-2,72
|
70
|
-3,48
|
45
|
-4,36
|
32
|
-6,24
|
18
|
-8,18
|
10
|
We have found our factors that have a sum of
10. Since our leading coefficient is
1, we can use
-8 and
18 directly as part of the factoring.
w2+10w−144=(w−8)(w+18)
Let's set each factor equal to zero and solve for
w.
(w−8)(w+18)=0
w−8=0w+18=0(I)(II)
w=8w+18=0
w=8w=-18
Since
w represents a measurement of distance, we can ignore the negative solution and say the width of the rectangle is
8 ft. To find the length, we can add
10 to the width and get
18 ft.