Solve x^2-2x=x-8

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

Solve for x (complex solution)

$$x=\frac{3+\sqrt{23}i}{2}\approx 1.5+2.397915762i$$

$$x=\frac{-\sqrt{23}i+3}{2}\approx 1.5-2.397915762i$$

Steps Using the Quadratic Formula

Subtract x from both sides.

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

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

$$x^{2}-3x=-8$$

Add 8 to both sides.

$$x^{2}-3x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 8}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 8}}{2}$$

Multiply -4 times 8.

$$x=\frac{-\left(-3\right)±\sqrt{9-32}}{2}$$

Add 9 to -32.

$$x=\frac{-\left(-3\right)±\sqrt{-23}}{2}$$

Take the square root of -23.

$$x=\frac{-\left(-3\right)±\sqrt{23}i}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{23}i}{2}$$

Now solve the equation x=\frac{3±\sqrt{23}i}{2} when ± is plus. Add 3 to i\sqrt{23}.

$$x=\frac{3+\sqrt{23}i}{2}$$

Now solve the equation x=\frac{3±\sqrt{23}i}{2} when ± is minus. Subtract i\sqrt{23} from 3.

$$x=\frac{-\sqrt{23}i+3}{2}$$

The equation is now solved.

$$x=\frac{3+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+3}{2}$$

Steps for Completing the Square

Subtract x from both sides.

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

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

$$x^{2}-3x=-8$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-8+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=-8+\frac{9}{4}$$

Add -8 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=-\frac{23}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=-\frac{23}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{23}i}{2} x-\frac{3}{2}=-\frac{\sqrt{23}i}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{3+\sqrt{23}i}{2} x=\frac{-\sqrt{23}i+3}{2}$$