-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-8=8-8

Subtract 8 from both sides of the equation.

-2x^{2}+x-8=0

Subtracting 8 from itself leaves 0.

x=\frac{-1±\sqrt{1^{2}-4\left(-2\right)\left(-8\right)}}{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)\left(-8\right)}}{2\left(-2\right)}

Square 1.

x=\frac{-1±\sqrt{1+8\left(-8\right)}}{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{-63}}{2\left(-2\right)}

Add 1 to -64.

x=\frac{-1±3\sqrt{7}i}{2\left(-2\right)}

Take the square root of -63.

x=\frac{-1±3\sqrt{7}i}{-4}

Multiply 2 times -2.

x=\frac{-1+3\sqrt{7}i}{-4}

Now solve the equation x=\frac{-1±3\sqrt{7}i}{-4} when ± is plus. Add -1 to 3i\sqrt{7}\approx 7.937253933i.

x=\frac{-3\sqrt{7}i+1}{4}

Divide -1+3i\sqrt{7}\approx -1+7.937253933i by -4.

x=\frac{-3\sqrt{7}i-1}{-4}

Now solve the equation x=\frac{-1±3\sqrt{7}i}{-4} when ± is minus. Subtract 3i\sqrt{7}\approx 7.937253933i from -1.

x=\frac{1+3\sqrt{7}i}{4}

Divide -1-3i\sqrt{7}\approx -1-7.937253933i by -4.

x=\frac{-3\sqrt{7}i+1}{4} x=\frac{1+3\sqrt{7}i}{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}=-0.5, the coefficient of the x term, by 2 to get -\frac{1}{4}=-0.25. Then add the square of -\frac{1}{4}=-0.25 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}=-0.25 by squaring both the numerator and the denominator of the fraction.

x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{63}{16}

Add -4 to \frac{1}{16}=0.0625.

\left(x-\frac{1}{4}\right)^{2}=-\frac{63}{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{63}{16}}

Take the square root of both sides of the equation.

x-\frac{1}{4}=\frac{3\sqrt{7}i}{4} x-\frac{1}{4}=-\frac{3\sqrt{7}i}{4}

Simplify.

x=\frac{1+3\sqrt{7}i}{4} x=\frac{-3\sqrt{7}i+1}{4}

Add \frac{1}{4}=0.25 to both sides of the equation.