Let s represent the length of one side of this rectangle, and w represent the width of the rectangle (length of one end). Then 2s + 2w = 12 units, and the area is a = s*w. Using the previous equation we can eliminate either s or w. Since we want s, eliminate w: 2w = 12 - 2s, or w = 6 - s.
Then the area of the rect. is a = s(6 - s) = 6s - s^2.
You have not shared the actual area of the rect., but have represented it by "a."
Suppose that the area is given and is 90. Then 90 square units = 6s - s^2, so that -s^2 + 6s - 90 = 0. Using the Pyth. Thm., we find s as follows:
-6 plus or minus sqrt( 6^2 - 4(-1)(-90) s = --------------------------------------------------- 2(-1)
Unfortunately, s would come out as a pair of complex numbers, which is not possible in a situation where you're calculating lengths.
Please recheck this problem. Was the aera of the rectangle, a, given?
