Solve x^2+x=3x-3

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

Solve for x (complex solution)

$$x=1+\sqrt{2}i\approx 1+1.414213562i$$

$$x=-\sqrt{2}i+1\approx 1-1.414213562i$$

Steps Using the Quadratic Formula

Subtract 3x from both sides.

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

Combine x and -3x to get -2x.

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

Add 3 to both sides.

$$x^{2}-2x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 3}}{2}$$

Multiply -4 times 3.

$$x=\frac{-\left(-2\right)±\sqrt{4-12}}{2}$$

Add 4 to -12.

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

Take the square root of -8.

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

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{2}i}{2}$$

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

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

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

$$x=1+\sqrt{2}i$$

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

Subtract 3x from both sides.

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

Combine x and -3x to get -2x.

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

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.

$$x^{2}-2x+1=-3+1$$

Add -3 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 1 to both sides of the equation.

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