Find the length and width of a rectangle whose perimeter is 40 feet and whose area is 96 square feet.

Let the length and width be l and w, respectively. Then, perimeter = 2 (l + w) = 40, and area = lw = 96. In other words:

l + w = 20

lw = 96

From here, we could guess and check to get that the length and width are 12 and 8, but let's do this rigorously:

By the first equation, l = 20 - w

Substituting into the second equation:

(20 - w) * w = 96

w^2 - 20w + 96 = 0

w^2 - 20w + 100 = 4

(w - 10) ^2 = 4

w - 10 = 2 or - 2

w = 12 or 8

The rest is trivial.