Solve x^2+4x-2x^2=8

$$x ^ { 2 } + 4 x - 2 x ^ { 2 } = 8$$

Solve for x (complex solution)

$$x=2-2i$$

$$x=2+2i$$

Steps Using the Quadratic Formula

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}+4x=8$$

Subtract 8 from both sides.

$$-x^{2}+4x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\left(-8\right)}}{2\left(-1\right)}$$

Multiply 4 times -8.

$$x=\frac{-4±\sqrt{16-32}}{2\left(-1\right)}$$

Add 16 to -32.

$$x=\frac{-4±\sqrt{-16}}{2\left(-1\right)}$$

Take the square root of -16.

$$x=\frac{-4±4i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±4i}{-2}$$

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

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

Divide -4+4i by -2.

$$x=2-2i$$

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

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

Divide -4-4i by -2.

$$x=2+2i$$

The equation is now solved.

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

Steps for Completing the Square

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}+4x=8$$

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=\frac{8}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=\frac{8}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=\frac{8}{-1}$$

Divide 8 by -1.

$$x^{2}-4x=-8$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=-8+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-8+4$$

Add -8 to 4.

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

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.

$$x-2=2i x-2=-2i$$

Add 2 to both sides of the equation.

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