Solve -1/2x^2=2x

$$- \frac { 1 } { 2 } x ^ { 2 } = 2 x$$

Solve for x

$$x=-4$$

$$x=0$$

Steps Using Factoring

Subtract 2x from both sides.

$$-\frac{1}{2}x^{2}-2x=0$$

Factor out x.

$$x\left(-\frac{1}{2}x-2\right)=0$$

To find equation solutions, solve x=0 and -\frac{x}{2}-2=0.

$$x=0 x=-4$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$-\frac{1}{2}x^{2}-2x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{2} for a, -2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

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

$$x=\frac{-\left(-2\right)±2}{2\left(-\frac{1}{2}\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±2}{2\left(-\frac{1}{2}\right)}$$

Multiply 2 times -\frac{1}{2}.

$$x=\frac{2±2}{-1}$$

Now solve the equation x=\frac{2±2}{-1} when ± is plus. Add 2 to 2.

$$x=\frac{4}{-1}$$

Divide 4 by -1.

$$x=-4$$

Now solve the equation x=\frac{2±2}{-1} when ± is minus. Subtract 2 from 2.

$$x=\frac{0}{-1}$$

Divide 0 by -1.

$$x=0$$

The equation is now solved.

$$x=-4 x=0$$

Steps for Completing the Square

Subtract 2x from both sides.

$$-\frac{1}{2}x^{2}-2x=0$$

Multiply both sides by -2.

$$\frac{-\frac{1}{2}x^{2}-2x}{-\frac{1}{2}}=\frac{0}{-\frac{1}{2}}$$

Dividing by -\frac{1}{2} undoes the multiplication by -\frac{1}{2}.

$$x^{2}+\left(-\frac{2}{-\frac{1}{2}}\right)x=\frac{0}{-\frac{1}{2}}$$

Divide -2 by -\frac{1}{2} by multiplying -2 by the reciprocal of -\frac{1}{2}.

$$x^{2}+4x=\frac{0}{-\frac{1}{2}}$$

Divide 0 by -\frac{1}{2} by multiplying 0 by the reciprocal of -\frac{1}{2}.

$$x^{2}+4x=0$$

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

$$x^{2}+4x+2^{2}=2^{2}$$

Square 2.

$$x^{2}+4x+4=4$$

Factor x^{2}+4x+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+2=2 x+2=-2$$

Subtract 2 from both sides of the equation.

$$x=0 x=-4$$