A rectangle has an area of 96 square feet and perimeter of 44 feet. What is length and width?

Perimeter=2L+2W=44, so L+W=22 and W=22-L. Area is 96=LW=L(22-L), so 22L-L^2=96 or L^2-22L+96=0, and (L-16)(L-6)=0. Therefore length and width are 6 and 16 feet.

I understand up to the L^2-22L+96=0. Do you just try all the factors to see which work, to get 16 and 6? Thanks!

Yes, but you can do it systematically be writing down all the factors of 96 in pairs: (1,96), (2,48), (3,32), (4,24), (6,16), (8,12). Then you write the corresponding sum of each pair of factors: (97), (50), (35), (28), (22), (20). You want the sum to be 22, the coefficient of L. So that's where (6,16) comes from. The sign of the 96 tells us that we need the sum of the factors. If the sign was minus, it would be the difference of the factors, so you'd list the differences instead. (Alternatively, use the quadratic formula, if you're not sure whether the quadratic is going to factorise.) After a bit of practice you get used to working out the factors quickly in your head.

Area A = 96

2l +2w =44...(1)

96 = wl.......(2)

Solving equation (1) and (2)

length l=16

width =6

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