Solve -2x^2=x+92x

$$- 2 x ^ { 2 } = x + 92 x$$

Solve for x

$$x = -\frac{93}{2} = -46\frac{1}{2} = -46.5$$

$$x=0$$

Steps Using Factoring

Combine x and 92x to get 93x.

$$-2x^{2}=93x$$

Subtract 93x from both sides.

$$-2x^{2}-93x=0$$

Factor out x.

$$x\left(-2x-93\right)=0$$

To find equation solutions, solve x=0 and -2x-93=0.

$$x=0 x=-\frac{93}{2}$$

Steps Using the Quadratic Formula

Combine x and 92x to get 93x.

$$-2x^{2}=93x$$

Subtract 93x from both sides.

$$-2x^{2}-93x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -93 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-93\right)±\sqrt{\left(-93\right)^{2}}}{2\left(-2\right)}$$

Take the square root of \left(-93\right)^{2}.

$$x=\frac{-\left(-93\right)±93}{2\left(-2\right)}$$

The opposite of -93 is 93.

$$x=\frac{93±93}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{93±93}{-4}$$

Now solve the equation x=\frac{93±93}{-4} when ± is plus. Add 93 to 93.

$$x=\frac{186}{-4}$$

Reduce the fraction \frac{186}{-4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{93}{2}$$

Now solve the equation x=\frac{93±93}{-4} when ± is minus. Subtract 93 from 93.

$$x=\frac{0}{-4}$$

Divide 0 by -4.

$$x=0$$

The equation is now solved.

$$x=-\frac{93}{2} x=0$$

Steps for Completing the Square

Combine x and 92x to get 93x.

$$-2x^{2}=93x$$

Subtract 93x from both sides.

$$-2x^{2}-93x=0$$

Divide both sides by -2.

$$\frac{-2x^{2}-93x}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{93}{-2}\right)x=\frac{0}{-2}$$

Divide -93 by -2.

$$x^{2}+\frac{93}{2}x=\frac{0}{-2}$$

Divide 0 by -2.

$$x^{2}+\frac{93}{2}x=0$$

Divide \frac{93}{2}, the coefficient of the x term, by 2 to get \frac{93}{4}. Then add the square of \frac{93}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{93}{2}x+\left(\frac{93}{4}\right)^{2}=\left(\frac{93}{4}\right)^{2}$$

Square \frac{93}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{93}{2}x+\frac{8649}{16}=\frac{8649}{16}$$

Factor x^{2}+\frac{93}{2}x+\frac{8649}{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}.

$$\left(x+\frac{93}{4}\right)^{2}=\frac{8649}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{93}{4}\right)^{2}}=\sqrt{\frac{8649}{16}}$$

Simplify.

$$x+\frac{93}{4}=\frac{93}{4} x+\frac{93}{4}=-\frac{93}{4}$$

Subtract \frac{93}{4} from both sides of the equation.

$$x=0 x=-\frac{93}{2}$$