SOLUTION: The length of a rectangle is 2feet less than 3times its width . Find the length and width if the area of the garden is 21 square feet

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Question 46952: The length of a rectangle is 2feet less than 3times its width . Find the length and width if the area of the garden is 21 square feet
Answer by aaaaaaaa(138) About Me  (Show Source):
You can put this solution on YOUR website!
l = length (in feet)
w = width (in feet)
We have that, by "The length of a rectangle is 2feet less than 3times its width":
l+=+3w+-+2
And we have that (since the area of a rectangle is calculated by multiplying length and width):
wl+=+21
Substituting l:
w%283w+-+2%29+=+21
3w%5E2+-+2w+=+21
3w%5E2+-+2w+-+21+=+0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aw%5E2%2Bbw%2Bc=0 (in our case 3w%5E2%2B-2w%2B-21+=+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=%28-2%29%5E2-4%2A3%2A-21=256.

Discriminant d=256 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+256+%29%29%2F2%5Ca.

w%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+256+%29%29%2F2%5C3+=+3
w%5B2%5D+=+%28-%28-2%29-sqrt%28+256+%29%29%2F2%5C3+=+-2.33333333333333

Quadratic expression 3w%5E2%2B-2w%2B-21 can be factored:
3w%5E2%2B-2w%2B-21+=+3%28w-3%29%2A%28w--2.33333333333333%29
Again, the answer is: 3, -2.33333333333333. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-2%2Ax%2B-21+%29


Since we only want positive solutions, w = 3, and we can say that (based on the first statement):
l+=+3%2A3+-+2
l+=+7
Verifying, we get that 7%2A3+=+21.