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

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Question 166951: x^2-x-1=0
Solve by completing the square.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E2-x-1 Start with the given expression.


Take half of the x coefficient -1 to get -1%2F2. In other words, %281%2F2%29%28-1%29=-1%2F2.


Now square -1%2F2 to get 1%2F4. In other words, %28-1%2F2%29%5E2=%28-1%2F2%29%28-1%2F2%29=1%2F4


x%5E2-x%2Bhighlight%281%2F4-1%2F4%29-1 Now add and subtract 1%2F4. Make sure to place this after the "x" term. Notice how 1%2F4-1%2F4=0. So the expression is not changed.


%28x%5E2-x%2B1%2F4%29-1%2F4-1 Group the first three terms.


%28x-1%2F2%29%5E2-1%2F4-1 Factor x%5E2-x%2B1%2F4 to get %28x-1%2F2%29%5E2.


%28x-1%2F2%29%5E2-5%2F4 Combine like terms.


So after completing the square, x%5E2-x-1 transforms to %28x-1%2F2%29%5E2-5%2F4. So x%5E2-x-1=%28x-1%2F2%29%5E2-5%2F4.


So x%5E2-x-1=0 is equivalent to %28x-1%2F2%29%5E2-5%2F4=0.



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%28x-1%2F2%29%5E2-5%2F4=0 Start with the given equation.


%28x-1%2F2%29%5E2=0%2B5%2F4Add 5%2F4 to both sides.


%28x-1%2F2%29%5E2=5%2F4 Combine like terms.


x-1%2F2=0%2B-sqrt%285%2F4%29 Take the square root of both sides.


x-1%2F2=sqrt%285%2F4%29 or x-1%2F2=-sqrt%285%2F4%29 Break up the "plus/minus" to form two equations.


x-1%2F2=sqrt%285%29%2F2 or x-1%2F2=-sqrt%285%29%2F2 Simplify the square root.


x=1%2F2%2Bsqrt%285%29%2F2 or x=1%2F2-sqrt%285%29%2F2 Add 1%2F2 to both sides.


x=%281%2Bsqrt%285%29%29%2F%282%29 or x=%281-sqrt%285%29%29%2F%282%29 Combine the fractions.


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Answer:


So the solutions are x=%281%2Bsqrt%285%29%29%2F%282%29 or x=%281-sqrt%285%29%29%2F%282%29.