A dog trainer has 64 ft of fencing that will be used to create a rectangular work area for dogs. If the trainer wants to enclose an area of 192 ft2, what will be the dimensions of the work area?

Answer 1

guessing way:
192 = 2^6*3 = 16*12 = 8*24
2(8+24) = 64
so the work area is 8x24

algebraic way:
If the pen's width is w, then the length is 32-w

w(32-w) = 192
w^2 - 32w + 192 = 0
(w-8)(w-24) = 0
w = 8 or 24
so, the pen is 8x24

Unless you use the quadratic formula, factoring the polynomial is essentially the same as guessing the right factors of 192!