SOLUTION: solve by completing the square: x^2-2x-1=0

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Question 116770: solve by completing the square: x^2-2x-1=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E2-2x-1=0 Start with the given equation


x%5E2-2x=1 Add 1 to both sides


Take half of the x coefficient -2 to get -1 (ie -2%2F2=-1)
Now square -1 to get 1 (ie %28-1%29%5E2=1)



x%5E2-2x%2B1=1%2B1 Add this result (1) to both sides. Now the expression x%5E2-2x%2B1 is a perfect square trinomial.




%28x-1%29%5E2=1%2B1 Factor x%5E2-2x%2B1 into %28x-1%29%5E2 (note: if you need help with factoring, check out this solver)



%28x-1%29%5E2=2 Combine like terms on the right side

x-1=0%2B-sqrt%282%29 Take the square root of both sides

x=1%2B-sqrt%282%29 Add 1 to both sides to isolate x.

So the expression breaks down to
x=1%2Bsqrt%282%29 or x=1-sqrt%282%29


So our answer is approximately
x=2.41421356237309 or x=-0.414213562373095

Here is visual proof

+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2-2x-1%29+ graph of y=x%5E2-2x-1


When we use the root finder feature on a calculator, we would find that the x-intercepts are x=2.41421356237309 and x=-0.414213562373095, so this verifies our answer.