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x^{2}=x
Cancel out 2 on both sides.
x^{2}-x=0
Subtract x from both sides.
x\left(x-1\right)=0
Factor out x.
x=0 x=1
To find equation solutions, solve x=0 and x-1=0.
x^{2}=x
Cancel out 2 on both sides.
x^{2}-x=0
Subtract x from both sides.
x=\frac{-\left(-1\right)±\sqrt{1}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±1}{2}
Take the square root of 1.
x=\frac{1±1}{2}
The opposite of -1 is 1.
x=\frac{2}{2}
Now solve the equation x=\frac{1±1}{2} when ± is plus. Add 1 to 1.
x=1
Divide 2 by 2.
x=\frac{0}{2}
Now solve the equation x=\frac{1±1}{2} when ± is minus. Subtract 1 from 1.
x=0
Divide 0 by 2.
x=1 x=0
The equation is now solved.
x^{2}=x
Cancel out 2 on both sides.
x^{2}-x=0
Subtract x from both sides.
x^{2}-x+\left(-\frac{1}{2}\right)^{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}=\frac{1}{4}
Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.
\left(x-\frac{1}{2}\right)^{2}=\frac{1}{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{1}{4}}
Take the square root of both sides of the equation.
x-\frac{1}{2}=\frac{1}{2} x-\frac{1}{2}=-\frac{1}{2}
Simplify.
x=1 x=0
Add \frac{1}{2} to both sides of the equation.