Answer:
12 in, 7 in
Step-by-step explanation:
The area of a rectangle is the product of length and width. Here, you are given the area, and an additional relation between length and width.
Setup
The two relations between length and width described by this problem are ...
A = LW . . . . . . . dimensions are of a rectangle
L = W +5 . . . . . . length is 5 inches more than width
A = 84 . . . . . . . . area in square inches
Solution
Substituting for L and A in the area formula, we have ...
84 = (W +5)(W)
We can solve this as a quadratic in any of several ways. One of those ways is by factoring.
Essentially, we're looking for factors of 84 that differ by 5. We can consider different factorizations of 84 to see what we get:
84 = 84×1 = 42×2 = 28×3 = 21×4 = 14×6 = 12×7
The differences between the factors in these pairs are 83, 40, 25, 17, 8, 5.
This means the last pair, with a difference of 5, is the one we're looking for.
W+5 = 12, W = 7
The rectangle is 12 inches long and 7 inches wide.
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Additional comment
As a quadratic in standard form, we would have ...
W² +5W -84 = 0 ⇒ (W +12)(W -7) = 0 ⇒ W = {7, -12}
If you were to solve this by completing the square, you would have ...
(W +2.5)² = 90.25 ⇒ W = -2.5 ±9.5 = {7, -12}
