Solve x^2+3x+5=x-2

$$x ^ { 2 } + 3 x + 5 = x - 2$$

Solve for x (complex solution)

$$x=-1+\sqrt{6}i\approx -1+2.449489743i$$

$$x=-\sqrt{6}i-1\approx -1-2.449489743i$$

Steps Using the Quadratic Formula

Subtract x from both sides.

$$x^{2}+3x+5-x=-2$$

Combine 3x and -x to get 2x.

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

Add 2 to both sides.

$$x^{2}+2x+5+2=0$$

Add 5 and 2 to get 7.

$$x^{2}+2x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 7}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 7}}{2}$$

Multiply -4 times 7.

$$x=\frac{-2±\sqrt{4-28}}{2}$$

Add 4 to -28.

$$x=\frac{-2±\sqrt{-24}}{2}$$

Take the square root of -24.

$$x=\frac{-2±2\sqrt{6}i}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{6}i}{2} when ± is plus. Add -2 to 2i\sqrt{6}.

$$x=\frac{-2+2\sqrt{6}i}{2}$$

Divide -2+2i\sqrt{6} by 2.

$$x=-1+\sqrt{6}i$$

Now solve the equation x=\frac{-2±2\sqrt{6}i}{2} when ± is minus. Subtract 2i\sqrt{6} from -2.

$$x=\frac{-2\sqrt{6}i-2}{2}$$

Divide -2-2i\sqrt{6} by 2.

$$x=-\sqrt{6}i-1$$

The equation is now solved.

$$x=-1+\sqrt{6}i x=-\sqrt{6}i-1$$

Steps for Completing the Square

Subtract x from both sides.

$$x^{2}+3x+5-x=-2$$

Combine 3x and -x to get 2x.

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

Subtract 5 from both sides.

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

Subtract 5 from -2 to get -7.

$$x^{2}+2x=-7$$

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.

$$x^{2}+2x+1^{2}=-7+1^{2}$$

Square 1.

$$x^{2}+2x+1=-7+1$$

Add -7 to 1.

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

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}.

$$\left(x+1\right)^{2}=-6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-6}$$

Simplify.

$$x+1=\sqrt{6}i x+1=-\sqrt{6}i$$

Subtract 1 from both sides of the equation.

$$x=-1+\sqrt{6}i x=-\sqrt{6}i-1$$