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x-x^{2}+2=0
Use the distributive property to multiply x by 1-x.
-x^{2}+x+2=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=1 ab=-2=-2
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+2. To find a and b, set up a system to be solved.
a=2 b=-1
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.
\left(-x^{2}+2x\right)+\left(-x+2\right)
Rewrite -x^{2}+x+2 as \left(-x^{2}+2x\right)+\left(-x+2\right).
-x\left(x-2\right)-\left(x-2\right)
Factor out -x in the first and -1 in the second group.
\left(x-2\right)\left(-x-1\right)
Factor out common term x-2 by using distributive property.
x=2 x=-1
To find equation solutions, solve x-2=0 and -x-1=0.
x-x^{2}+2=0
Use the distributive property to multiply x by 1-x.
-x^{2}+x+2=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{-1±\sqrt{1^{2}-4\left(-1\right)\times 2}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-1\right)\times 2}}{2\left(-1\right)}
Square 1.
x=\frac{-1±\sqrt{1+4\times 2}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-1±\sqrt{1+8}}{2\left(-1\right)}
Multiply 4 times 2.
x=\frac{-1±\sqrt{9}}{2\left(-1\right)}
Add 1 to 8.
x=\frac{-1±3}{2\left(-1\right)}
Take the square root of 9.
x=\frac{-1±3}{-2}
Multiply 2 times -1.
x=\frac{2}{-2}
Now solve the equation x=\frac{-1±3}{-2} when ± is plus. Add -1 to 3.
x=-1
Divide 2 by -2.
x=-\frac{4}{-2}
Now solve the equation x=\frac{-1±3}{-2} when ± is minus. Subtract 3 from -1.
x=2
Divide -4 by -2.
x=-1 x=2
The equation is now solved.
x-x^{2}+2=0
Use the distributive property to multiply x by 1-x.
x-x^{2}=-2
Subtract 2 from both sides. Anything subtracted from zero gives its negation.
-x^{2}+x=-2
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}+x}{-1}=-\frac{2}{-1}
Divide both sides by -1.
x^{2}+\frac{1}{-1}x=-\frac{2}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-x=-\frac{2}{-1}
Divide 1 by -1.
x^{2}-x=2
Divide -2 by -1.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=2+\left(-\frac{1}{2}\right)^{2}
Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-x+\frac{1}{4}=2+\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
x^{2}-x+\frac{1}{4}=\frac{9}{4}
Add 2 to \frac{1}{4}.
\left(x-\frac{1}{2}\right)^{2}=\frac{9}{4}
Factor x^{2}-x+\frac{1}{4}. 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-\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{3}{2} x-\frac{1}{2}=-\frac{3}{2}
Simplify.
x=2 x=-1
Add \frac{1}{2} to both sides of the equation.