Solve 2x=-2x^2-2

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

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{3}i}{2}\approx -0.5+0.866025404i$$

$$x=\frac{-\sqrt{3}i-1}{2}\approx -0.5-0.866025404i$$

Steps Using the Quadratic Formula

Add 2x^{2} to both sides.

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

Add 2 to both sides.

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

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.

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

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

$$x=\frac{-2±\sqrt{2^{2}-4\times 2\times 2}}{2\times 2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 2\times 2}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-2±\sqrt{4-8\times 2}}{2\times 2}$$

Multiply -8 times 2.

$$x=\frac{-2±\sqrt{4-16}}{2\times 2}$$

Add 4 to -16.

$$x=\frac{-2±\sqrt{-12}}{2\times 2}$$

Take the square root of -12.

$$x=\frac{-2±2\sqrt{3}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-2±2\sqrt{3}i}{4}$$

Now solve the equation x=\frac{-2±2\sqrt{3}i}{4} when ± is plus. Add -2 to 2i\sqrt{3}.

$$x=\frac{-2+2\sqrt{3}i}{4}$$

Divide -2+2i\sqrt{3} by 4.

$$x=\frac{-1+\sqrt{3}i}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{3}i}{4} when ± is minus. Subtract 2i\sqrt{3} from -2.

$$x=\frac{-2\sqrt{3}i-2}{4}$$

Divide -2-2i\sqrt{3} by 4.

$$x=\frac{-\sqrt{3}i-1}{2}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i-1}{2}$$

Steps for Completing the Square

Add 2x^{2} to both sides.

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

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.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 2 by 2.

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

Divide -2 by 2.

$$x^{2}+x=-1$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add -1 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=-\frac{3}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=-\frac{3}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{3}i}{2} x+\frac{1}{2}=-\frac{\sqrt{3}i}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{-1+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i-1}{2}$$