Solve 12x^2-22x=x^2-12

$$12 x ^ { 2 } - 22 x = x ^ { 2 } - 12$$

Solve for x (complex solution)

$$x=\frac{\sqrt{11}i}{11}+1\approx 1+0.301511345i$$

$$x=-\frac{\sqrt{11}i}{11}+1\approx 1-0.301511345i$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$12x^{2}-22x-x^{2}=-12$$

Combine 12x^{2} and -x^{2} to get 11x^{2}.

$$11x^{2}-22x=-12$$

Add 12 to both sides.

$$11x^{2}-22x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 11 for a, -22 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 11\times 12}}{2\times 11}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 11\times 12}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-22\right)±\sqrt{484-44\times 12}}{2\times 11}$$

Multiply -44 times 12.

$$x=\frac{-\left(-22\right)±\sqrt{484-528}}{2\times 11}$$

Add 484 to -528.

$$x=\frac{-\left(-22\right)±\sqrt{-44}}{2\times 11}$$

Take the square root of -44.

$$x=\frac{-\left(-22\right)±2\sqrt{11}i}{2\times 11}$$

The opposite of -22 is 22.

$$x=\frac{22±2\sqrt{11}i}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{22±2\sqrt{11}i}{22}$$

Now solve the equation x=\frac{22±2\sqrt{11}i}{22} when ± is plus. Add 22 to 2i\sqrt{11}.

$$x=\frac{22+2\sqrt{11}i}{22}$$

Divide 22+2i\sqrt{11} by 22.

$$x=\frac{\sqrt{11}i}{11}+1$$

Now solve the equation x=\frac{22±2\sqrt{11}i}{22} when ± is minus. Subtract 2i\sqrt{11} from 22.

$$x=\frac{-2\sqrt{11}i+22}{22}$$

Divide 22-2i\sqrt{11} by 22.

$$x=-\frac{\sqrt{11}i}{11}+1$$

The equation is now solved.

$$x=\frac{\sqrt{11}i}{11}+1 x=-\frac{\sqrt{11}i}{11}+1$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$12x^{2}-22x-x^{2}=-12$$

Combine 12x^{2} and -x^{2} to get 11x^{2}.

$$11x^{2}-22x=-12$$

Divide both sides by 11.

$$\frac{11x^{2}-22x}{11}=-\frac{12}{11}$$

Dividing by 11 undoes the multiplication by 11.

$$x^{2}+\left(-\frac{22}{11}\right)x=-\frac{12}{11}$$

Divide -22 by 11.

$$x^{2}-2x=-\frac{12}{11}$$

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.

$$x^{2}-2x+1=-\frac{12}{11}+1$$

Add -\frac{12}{11} to 1.

$$x^{2}-2x+1=-\frac{1}{11}$$

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}.

$$\left(x-1\right)^{2}=-\frac{1}{11}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-\frac{1}{11}}$$

Simplify.

$$x-1=\frac{\sqrt{11}i}{11} x-1=-\frac{\sqrt{11}i}{11}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{11}i}{11}+1 x=-\frac{\sqrt{11}i}{11}+1$$