A builder cuts a rectangular hole in a ceiling to install an exhaust fan. The width is 3 in. less than the length and the area is 180 in². Find the length and width of the hole.

Area of a rectangle = l * w

A = Area of a rectangle

l = length

w = width

In our problem,

l = x

w = x - 3

A = 180 square inches

Plug our numbers into the area formula of a rectangle

180 square inches = x (x-3)

Distribute the x into (x-3)

180 square inches = x^2 - 3x

subtract 180 from both sides

0 = x^2 - 3x - 180.

Factor the express on the left side of the equation

0 = (x-15) (x+12).

Set each term equal to zero and solve for x

x+12 = 0.

Subtract 12 from both sides

x = - 12 <-- - lengths of a rectangle cannot be negative, therefore, we need to check the other term we factored.

set x-15 equal to zero

x-15 = 0.

Add 15 to both sides

x = 15.

We know l = x so now we know that the length i = 15 inches.

We also know that the width = length - 3

w = 15 - 3

w = 12

The width is 12 inches and the length is 15 inches