What is the width of the rectangle if the length is 4 times the reciprocal of the width and the perimeter is 10?

Create formulas for length and perimeter.

L = 4/W
10 = 2*L + 2*W

Substitute formula for length for the length in perimeter formula
10 = 2*4/W + 2*W
10 = 8/W + 2*W
5 = 4/W + W
5W = 4 + W^2
0 = 4 - 5W + W^2
W^2 -5W + 4 = 0
You now have a standard quadratic equation with A=1, B=-5, and C=4
Using the quadratic formula you get two possible answers
W = 1, or W = 4
Let's check them
Width = 1, L = 4/1 = 4  
Width = 4, L = 4/4 = 1
Both make the same 1x4 rectangle. And the perimeter is 2*1 + 2*4 = 2+8 = 10