SOLUTION: A rectangular piece of picture matting is 2 inches longer than it is wide. A rectangular region is cut out of the middle for a picture, leaving 1 inch of matting all around. If the

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Question 282144: A rectangular piece of picture matting is 2 inches longer than it is wide. A rectangular region is cut out of the middle for a picture, leaving 1 inch of matting all around. If the area of the region that is removed is 24 square inches, what are the dimensions of the matting?
Answer by Deina(147) About Me  (Show Source):
You can put this solution on YOUR website!
First we will find the dimensions of the piece removed.
The formula for the area of a rectangle is:
L * W = 24
L = W + 2

Substitue the value of L for L
(W + 2) * W = 24

Expand:
W^2 + 2W = 24

Subtract 24 from each side:
W^2 + 2W - 24 = 24 - 24

Solve the quadratic equation:
W^2 + 2W - 24 = 0

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aw%5E2%2Bbw%2Bc=0 (in our case 1w%5E2%2B2w%2B-24+=+0) has the following solutons:

w%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%282%29%5E2-4%2A1%2A-24=100.

Discriminant d=100 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+100+%29%29%2F2%5Ca.

w%5B1%5D+=+%28-%282%29%2Bsqrt%28+100+%29%29%2F2%5C1+=+4
w%5B2%5D+=+%28-%282%29-sqrt%28+100+%29%29%2F2%5C1+=+-6

Quadratic expression 1w%5E2%2B2w%2B-24 can be factored:
1w%5E2%2B2w%2B-24+=+1%28w-4%29%2A%28w--6%29
Again, the answer is: 4, -6. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B-24+%29


Since -6 isn't a reasonable answer considering our problem, the width of the inner rectangle is 4" and the length is 6".
W = 4, L = 6

Since there was 1" margin on all four sides we add 2" to W and L to get the dimensions of the matting
W + 2 = 6, L + 2 = 8.

The overall dimensions of the matting are 6" wide by 8" long and bob's your uncle!