SOLUTION: Please help me to find the answer. Find the length of a side of an equilateral triangle whose altitude is 3 feet shorter than a side.

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Question 843443: Please help me to find the answer. Find the length of a side of an equilateral triangle whose altitude is 3 feet shorter than a side.
Answer by LinnW(1048) About Me  (Show Source):
You can put this solution on YOUR website!
Set s = length of a side.
Bisecting one of the angles produces two triangles,
each with a base of (1/2)s. Since the height is 3
less than a side length, height = s - 3
By Pythagorean theorem
%28%281%2F2%29s%29%5E2+%2B+%28s+-+3%29%5E2+=+s%5E2
%281%2F4%29s%5E2+%2B+s%5E2+-6s+%2B9+=+s%5E2
%281%2F4%29s%5E2+%2B+s%5E2+-6s+%2B9+=+s%5E2
%285%2F4%29s%5E2+-6s+%2B9+=+s%5E2
multiply each side by 4
5s%5E2+-24s+%2B36+=+4s%5E2
subtract 4s^2 from each side
s%5E2+-24s+%2B36+=+0
There are two possible values for s.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-24x%2B36+=+0) has the following solutons:

x%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=%28-24%29%5E2-4%2A1%2A36=432.

Discriminant d=432 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--24%2B-sqrt%28+432+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-24%29%2Bsqrt%28+432+%29%29%2F2%5C1+=+22.3923048454133
x%5B2%5D+=+%28-%28-24%29-sqrt%28+432+%29%29%2F2%5C1+=+1.60769515458674

Quadratic expression 1x%5E2%2B-24x%2B36 can be factored:
1x%5E2%2B-24x%2B36+=+1%28x-22.3923048454133%29%2A%28x-1.60769515458674%29
Again, the answer is: 22.3923048454133, 1.60769515458674. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-24%2Ax%2B36+%29