Cynthia Besch wants to buy a rug for a room that is 20 ft wide and 26 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 432 square feet of carpeting. What dimensions should the rug​ have?

Let x represent the width of the "uniform strip."

Write formulas:  one for the length of the rug, one for the width.

If the room is 20 ft wide, the rug width would have to be 20-2x, and the run length would be 26-2x.

Rug area is 432 sq ft, and this equals (rug length)(rug width), or

432 = (26-2x)(20-2x)  = 520 -40x  - 52x  + 4x^2   and this is 432.

Subtr. 432 from both sides:    520 - 432 - 92x + 4x^2 = 0.  This is a quadratic equation that could be solved in various ways.

4x^2 - 92x + 88 = 0.  Let's reduce this by div. all terms by 4:

x^2 -23x + 22 = 0

Easily factored!  Note that -1x and -22x add up to -23x, and that (-1)(-12) = 22.  Thus, the factors are (x-1)(x-22) = 0, so that x = 1 or x = 22.
Remembering that x represents the strip width, we eliminate x = 22 and keep x = 1.

The rug dimensions should be (26-2) by (20-2), or 24 by 18 feet.

Check:  Does this area come out to 432 sq ft?   (24)(18) = 432/  YES