SOLUTION: x^2=5x-1 Solve by any method

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Question 87585: x^2=5x-1
Solve by any method
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2=5x-1

x%5E2-5x%2B1=0 Get everything to one side

Now let's use the quadratic formula to solve for x:
Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general form of the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve x%5E2-5%2Ax%2B1=0

x+=+%285+%2B-+sqrt%28+%28-5%29%5E2-4%2A1%2A1+%29%29%2F%282%2A1%29 Plug in a=1, b=-5, and c=1



x+=+%285+%2B-+sqrt%28+25-4%2A1%2A1+%29%29%2F%282%2A1%29 Square -5 to get 25



x+=+%285+%2B-+sqrt%28+25%2B-4+%29%29%2F%282%2A1%29 Multiply -4%2A1%2A1 to get -4



x+=+%285+%2B-+sqrt%28+21+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%285+%2B-+sqrt%2821%29%29%2F%282%2A1%29 Simplify the square root



x+=+%285+%2B-+sqrt%2821%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%285+%2B+sqrt%2821%29%29%2F2 or x+=+%285+-+sqrt%2821%29%29%2F2


Which approximate to

x=4.79128784747792 or x=0.20871215252208


So our solutions are:
x=4.79128784747792 or x=0.20871215252208

Notice when we graph x%5E2-5%2Ax%2B1 we get:



when we use the root finder feature on a calculator, we find that x=4.79128784747792 and x=0.20871215252208.So this verifies our answer