If a rug is to fit in a room so that a boarder of even width is left on all four sides. if the roo is 12 feet by 15 feet and the area of the rug is 108 square feet, how wide will the boarder be

Define x:

Let x be the width of the border

Length of the rug = 15 -2x
Width of the rug = 12 -2x 

Area of the rug = (15 - 2x)(12 - 2x) 

Construct equation:

Given that the area of the rug os 108 ft²

(15 - 2x)(12 - 2x)  = 108
Apply distributive property:
180 - 20x - 24x + 4x² = 108

Collect all terms on the left hand side:
4x² - 44x + 72 = 0

Factorise:
x² - 11x + 18 = 0

(x - 2) (x - 9)

Apply zero product property:
x = 2 or x = 9

Check x:

When x = 9, 
Width = 12 - 9 - 9 = -6
⇒ x = 9 is rejected since width cannot be negative

When x = 2
Length = 15 - 2 - 2 = 11 ft
Width = 12 - 2 - 2 = 8 ft 

Answer: The border is 2 feet wide.