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-x^{2}+2x=\frac{1}{4}
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^{2}+2x-\frac{1}{4}=\frac{1}{4}-\frac{1}{4}
Subtract \frac{1}{4} from both sides of the equation.
-x^{2}+2x-\frac{1}{4}=0
Subtracting \frac{1}{4} from itself leaves 0.
x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\left(-\frac{1}{4}\right)}}{2\left(-1\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and -\frac{1}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-2±\sqrt{4-4\left(-1\right)\left(-\frac{1}{4}\right)}}{2\left(-1\right)}
Square 2.
x=\frac{-2±\sqrt{4+4\left(-\frac{1}{4}\right)}}{2\left(-1\right)}
Multiply -4 times -1.
x=\frac{-2±\sqrt{4-1}}{2\left(-1\right)}
Multiply 4 times -\frac{1}{4}.
x=\frac{-2±\sqrt{3}}{2\left(-1\right)}
Add 4 to -1.
x=\frac{-2±\sqrt{3}}{-2}
Multiply 2 times -1.
x=\frac{\sqrt{3}-2}{-2}
Now solve the equation x=\frac{-2±\sqrt{3}}{-2} when ± is plus. Add -2 to \sqrt{3}.
x=-\frac{\sqrt{3}}{2}+1
Divide -2+\sqrt{3} by -2.
x=\frac{-\sqrt{3}-2}{-2}
Now solve the equation x=\frac{-2±\sqrt{3}}{-2} when ± is minus. Subtract \sqrt{3} from -2.
x=\frac{\sqrt{3}}{2}+1
Divide -2-\sqrt{3} by -2.
x=-\frac{\sqrt{3}}{2}+1 x=\frac{\sqrt{3}}{2}+1
The equation is now solved.
-x^{2}+2x=\frac{1}{4}
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{\frac{1}{4}}{-1}
Divide both sides by -1.
x^{2}+\frac{2}{-1}x=\frac{\frac{1}{4}}{-1}
Dividing by -1 undoes the multiplication by -1.
x^{2}-2x=\frac{\frac{1}{4}}{-1}
Divide 2 by -1.
x^{2}-2x=-\frac{1}{4}
Divide \frac{1}{4} by -1.
x^{2}-2x+1=-\frac{1}{4}+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=\frac{3}{4}
Add -\frac{1}{4} to 1.
\left(x-1\right)^{2}=\frac{3}{4}
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{\frac{3}{4}}
Take the square root of both sides of the equation.
x-1=\frac{\sqrt{3}}{2} x-1=-\frac{\sqrt{3}}{2}
Simplify.
x=\frac{\sqrt{3}}{2}+1 x=-\frac{\sqrt{3}}{2}+1
Add 1 to both sides of the equation.