Solve x^2+4x+6=2x

$${ x }^{ 2 } +4x+6=2x$$

Solve for x (complex solution)

$$x=-1+\sqrt{5}i\approx -1+2.236067977i$$

$$x=-\sqrt{5}i-1\approx -1-2.236067977i$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

$$x^{2}+4x+6-2x=0$$

Combine 4x and -2x to get 2x.

$$x^{2}+2x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 6}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-2±\sqrt{4-24}}{2}$$

Add 4 to -24.

$$x=\frac{-2±\sqrt{-20}}{2}$$

Take the square root of -20.

$$x=\frac{-2±2\sqrt{5}i}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{5}i}{2} when ± is plus. Add -2 to 2i\sqrt{5}.

$$x=\frac{-2+2\sqrt{5}i}{2}$$

Divide -2+2i\sqrt{5} by 2.

$$x=-1+\sqrt{5}i$$

Now solve the equation x=\frac{-2±2\sqrt{5}i}{2} when ± is minus. Subtract 2i\sqrt{5} from -2.

$$x=\frac{-2\sqrt{5}i-2}{2}$$

Divide -2-2i\sqrt{5} by 2.

$$x=-\sqrt{5}i-1$$

The equation is now solved.

$$x=-1+\sqrt{5}i x=-\sqrt{5}i-1$$

Steps for Completing the Square

Subtract 2x from both sides.

$$x^{2}+4x+6-2x=0$$

Combine 4x and -2x to get 2x.

$$x^{2}+2x+6=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+2x=-6$$

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

$$x^{2}+2x+1^{2}=-6+1^{2}$$

Square 1.

$$x^{2}+2x+1=-6+1$$

Add -6 to 1.

$$x^{2}+2x+1=-5$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=-5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-5}$$

Simplify.

$$x+1=\sqrt{5}i x+1=-\sqrt{5}i$$

Subtract 1 from both sides of the equation.

$$x=-1+\sqrt{5}i x=-\sqrt{5}i-1$$