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x-x^{2}=-x-1
Subtract x^{2} from both sides.
x-x^{2}+x=-1
Add x to both sides.
2x-x^{2}=-1
Combine x and x to get 2x.
2x-x^{2}+1=0
Add 1 to both sides.
-x^{2}+2x+1=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-1\right)}}{2\left(-1\right)}
Square 2.
x=\frac{-2±\sqrt{4+4}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-2±\sqrt{8}}{2\left(-1\right)}
Add 4 to 4.
x=\frac{-2±2\sqrt{2}}{2\left(-1\right)}
Take the square root of 8.
x=\frac{-2±2\sqrt{2}}{-2}
Multiply 2 times -1.
x=\frac{2\sqrt{2}-2}{-2}
Now solve the equation x=\frac{-2±2\sqrt{2}}{-2} when ± is plus. Add -2 to 2\sqrt{2}.
x=1-\sqrt{2}
Divide -2+2\sqrt{2} by -2.
x=\frac{-2\sqrt{2}-2}{-2}
Now solve the equation x=\frac{-2±2\sqrt{2}}{-2} when ± is minus. Subtract 2\sqrt{2} from -2.
x=\sqrt{2}+1
Divide -2-2\sqrt{2} by -2.
x=1-\sqrt{2} x=\sqrt{2}+1
The equation is now solved.
x-x^{2}=-x-1
Subtract x^{2} from both sides.
x-x^{2}+x=-1
Add x to both sides.
2x-x^{2}=-1
Combine x and x to get 2x.
-x^{2}+2x=-1
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{-x^{2}+2x}{-1}=-\frac{1}{-1}
Divide both sides by -1.
x^{2}+\frac{2}{-1}x=-\frac{1}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-2x=-\frac{1}{-1}
Divide 2 by -1.
x^{2}-2x=1
Divide -1 by -1.
x^{2}-2x+1=1+1
Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-2x+1=2
Add 1 to 1.
\left(x-1\right)^{2}=2
Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-1\right)^{2}}=\sqrt{2}
Take the square root of both sides of the equation.
x-1=\sqrt{2} x-1=-\sqrt{2}
Simplify.
x=\sqrt{2}+1 x=1-\sqrt{2}
Add 1 to both sides of the equation.