SOLUTION: the three sides of a right-angled triangle measure x-1, x+6, and 2x+1 in length. what are the possible lengths of the hypotenuse?

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Question 951221: the three sides of a right-angled triangle measure x-1, x+6, and 2x+1 in length.
what are the possible lengths of the hypotenuse?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The hypotenuse can be either x%2B6 or 2x%2B1.
So,
%28x-1%29%5E2%2B%28x%2B6%29%5E2=%282x%2B1%29%5E2
x%5E2-2x%2B1%2Bx%5E2%2B12x%2B36=4x%5E2%2B4x%2B1
2x%5E2%2B10x%2B37=4x%5E2%2B4x%2B1
2x%5E2-6x-36=0
x%5E2-3x-18=0
%28x-6%29%28x%2B3%29=0
Only the positive solution makes sense here.
x-6=0
x=6
So then, 2x%2B1=2%286%29%2B1=12%2B1=highlight%2813%29
and
%28x-1%29%5E2%2B%282x%2B1%29%5E2=%28x-6%29%5E2
x%5E2-2x%2B1%2B4x%5E2%2B4x%2B1=x%5E2-12x%2B36
5x%5E2%2B2x%2B2=x%5E2-12x%2B36
4x%5E2%2B14x-34=0
2x%5E2%2B7x-17=0
2%28x%5E2%2B%287%2F2%29x%29=17
2%28x%5E2%2B%287%2F2%29x%2B%287%2F4%29%5E2%29=17%2B2%287%2F4%29%5E2
2%28x%2B7%2F4%29%5E2=136%2F8%2B49%2F8
2%28x%2B7%2F4%29%5E2=185%2F8
%28x%2B7%2F4%29%5E2=185%2F16
x%2B7%2F4=0+%2B-+sqrt%28185%29%2F4
x=-7%2F4+%2B-+sqrt%28185%29%2F4
Again use only the positive solution.
x=-7%2F4%2Bsqrt%28185%29%2F4
So then,
x%2B6=24%2F4-7%2F4%2Bsqrt%28185%29%2F4
x%2B6=highlight%2817%2F4%2Bsqrt%28185%29%2F4%29