SOLUTION: x^-x=1 what is the solution set{ }? quadratic formula.

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Question 236187: x^-x=1 what is the solution set{ }?
quadratic formula.
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 equation.


x%5E2-x-1=0 Subtract 1 from both sides.


Notice that the quadratic x%5E2-x-1 is in the form of Ax%5E2%2BBx%2BC where A=1, B=-1, and C=-1


Let's use the quadratic formula to solve for "x":


x+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29 Start with the quadratic formula


x+=+%28-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-1%29+%29%29%2F%282%281%29%29 Plug in A=1, B=-1, and C=-1


x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-1%29+%29%29%2F%282%281%29%29 Negate -1 to get 1.


x+=+%281+%2B-+sqrt%28+1-4%281%29%28-1%29+%29%29%2F%282%281%29%29 Square -1 to get 1.


x+=+%281+%2B-+sqrt%28+1--4+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-1%29 to get -4


x+=+%281+%2B-+sqrt%28+1%2B4+%29%29%2F%282%281%29%29 Rewrite sqrt%281--4%29 as sqrt%281%2B4%29


x+=+%281+%2B-+sqrt%28+5+%29%29%2F%282%281%29%29 Add 1 to 4 to get 5


x+=+%281+%2B-+sqrt%28+5+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%281%2Bsqrt%285%29%29%2F%282%29 or x+=+%281-sqrt%285%29%29%2F%282%29 Break up the expression.


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


which approximate to x=1.618 or x=-0.618