SOLUTION: the length of the rectangular garden is one more than twice its width. Its area is 80m^2. Express the length and width of the rectangular garden in terms of x. write the quadratic

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Question 884403: the length of the rectangular garden is one more than twice its width. Its area is 80m^2. Express the length and width of the rectangular garden in terms of x. write the quadratic equation. find the lenght of the diagonal path across the garden.
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
x may be any one of three different quantities. I will use w for width and L for length. No use of x.

The first sentence becomes L=1%2B2w.
Area is wL=w%282w%2B1%29=80.

2w%5E2%2Bw=80
2w%5E2%2B2w-80=0
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Discriminant, 2%5E2-4%2A2%2A%28-80%29=4%2B8%2A80=644
sqrt%28644%29=sqrt%284%2A161%29=sqrt%284%2A7%2A23%29=2sqrt%28161%29
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w=%28-2%2B-+2sqrt%28161%29%29%2F4
You must use the plus square root in this case.
w=%28-2%2B2sqrt%28161%29%29%2F4
highlight%28w=%28-1%2Bsqrt%28161%29%29%2F2%29

Use solution for w to find solution for L.
L=1%2B2w
L=1%2B2%28-1%2Bsqrt%28161%29%29%2F2
L=1%2B%28-1%2Bsqrt%28161%29%29
highlight%28L=sqrt%28161%29%29

You can go further and and use Pythagorean Theorem to get the diagonal:
Let r = the diagonal,
r%5E2=L%5E2%2Bw%5E2
r%5E2=161%2B%28%28-1%2Bsqrt%28161%29%29%2F2%29%5E2
r%5E2=161%2B%28%28sqrt%28161%29-1%29%2F2%29%5E2
r%5E2=161%2B%28sqrt%28161%29-1%29%5E2%2F4
r%5E2=%28644%2B%28sqrt%28161%29-1%29%5E2%29%2F4
%28644%2B161-2sqrt%28161%29%2B1%29%2F4
%28644%2B162-2sqrt%28161%29%29%2F4
%28806-2sqrt%28161%29%29%2F4
r%5E2=%28806-2sqrt%28161%29%29%2F4
r=sqrt%28%28806-2sqrt%28161%29%29%2F4%29
highlight%28r=sqrt%28806-2sqrt%28161%29%29%2F2%29, the diagonal across the garden.