SOLUTION: I appreciate a respond "If {{{x+1/x=8}}}, what is the value of {{{x^4+1/x^4}}}? "

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Question 201440: I appreciate a respond "If x%2B1%2Fx=8, what is the value of x%5E4%2B1%2Fx%5E4? "

Answer by Alan3354(8275) About Me  (Show Source):
You can put this solution on YOUR website!
"If x%2B1%2Fx=8, what is the value of x%5E4%2B1%2Fx%5E4? "
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x%2B1%2Fx=8 Multiply thru by x
x%5E2+%2B+1+=+8x
x%5E2+-+8x+%2B+1+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-8x%2B1+=+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=%28-8%29%5E2-4%2A1%2A1=60.

Discriminant d=60 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--8%2B-sqrt%28+60+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-8%29%2Bsqrt%28+60+%29%29%2F2%5C1+=+7.87298334620742
x%5B2%5D+=+%28-%28-8%29-sqrt%28+60+%29%29%2F2%5C1+=+0.127016653792583

Quadratic expression 1x%5E2%2B-8x%2B1 can be factored:
1x%5E2%2B-8x%2B1+=+%28x-7.87298334620742%29%2A%28x-0.127016653792583%29
Again, the answer is: 7.87298334620742, 0.127016653792583. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-8%2Ax%2B1+%29

x+=+4%2Bsqrt%2815%29
x%5E2+=+31+%2B+8sqrt%2815%29
x%5E4+=+1921+%2B+496sqrt%2815%29
x%5E4%2B1%2Fx%5E4 = %281922%2B496sqrt%2815%29%29%2F%281921%2B496sqrt%2815%29%29
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x+=+4-sqrt%2815%29
x%5E2+=+1+-+8sqrt%2815%29
x%5E4+=+961+-+16sqrt%2815%29
x%5E4%2B1%2Fx%5E4 = %28962-16sqrt%2815%29%29%2F%28961-16sqrt%2815%29%29