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-2x^{2}+x=-8
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.
-2x^{2}+x-\left(-8\right)=-8-\left(-8\right)
Add 8 to both sides of the equation.
-2x^{2}+x-\left(-8\right)=0
Subtracting -8 from itself leaves 0.
-2x^{2}+x+8=0
Subtract -8 from 0.
x=\frac{-1±\sqrt{1^{2}-4\left(-2\right)\times 8}}{2\left(-2\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 1 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-2\right)\times 8}}{2\left(-2\right)}
Square 1.
x=\frac{-1±\sqrt{1+8\times 8}}{2\left(-2\right)}
Multiply -4 times -2.
x=\frac{-1±\sqrt{1+64}}{2\left(-2\right)}
Multiply 8 times 8.
x=\frac{-1±\sqrt{65}}{2\left(-2\right)}
Add 1 to 64.
x=\frac{-1±\sqrt{65}}{-4}
Multiply 2 times -2.
x=\frac{\sqrt{65}-1}{-4}
Now solve the equation x=\frac{-1±\sqrt{65}}{-4} when ± is plus. Add -1 to \sqrt{65}.
x=\frac{1-\sqrt{65}}{4}
Divide -1+\sqrt{65} by -4.
x=\frac{-\sqrt{65}-1}{-4}
Now solve the equation x=\frac{-1±\sqrt{65}}{-4} when ± is minus. Subtract \sqrt{65} from -1.
x=\frac{\sqrt{65}+1}{4}
Divide -1-\sqrt{65} by -4.
x=\frac{1-\sqrt{65}}{4} x=\frac{\sqrt{65}+1}{4}
The equation is now solved.
-2x^{2}+x=-8
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{-2x^{2}+x}{-2}=-\frac{8}{-2}
Divide both sides by -2.
x^{2}+\frac{1}{-2}x=-\frac{8}{-2}
Dividing by -2 undoes the multiplication by -2.
x^{2}-\frac{1}{2}x=-\frac{8}{-2}
Divide 1 by -2.
x^{2}-\frac{1}{2}x=4
Divide -8 by -2.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=4+\left(-\frac{1}{4}\right)^{2}
Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{1}{2}x+\frac{1}{16}=4+\frac{1}{16}
Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{65}{16}
Add 4 to \frac{1}{16}.
\left(x-\frac{1}{4}\right)^{2}=\frac{65}{16}
Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. 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}{4}\right)^{2}}=\sqrt{\frac{65}{16}}
Take the square root of both sides of the equation.
x-\frac{1}{4}=\frac{\sqrt{65}}{4} x-\frac{1}{4}=-\frac{\sqrt{65}}{4}
Simplify.
x=\frac{\sqrt{65}+1}{4} x=\frac{1-\sqrt{65}}{4}
Add \frac{1}{4} to both sides of the equation.