Solve x=6/x-6

$$x= \frac{ 6 }{ x-6 }$$

Solve for x

$$x=\sqrt{15}+3\approx 6.872983346$$

$$x=3-\sqrt{15}\approx -0.872983346$$

Steps Using the Quadratic Formula

Subtract \frac{6}{x-6} from both sides.

$$x-\frac{6}{x-6}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-6}{x-6}.

$$\frac{x\left(x-6\right)}{x-6}-\frac{6}{x-6}=0$$

Since \frac{x\left(x-6\right)}{x-6} and \frac{6}{x-6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x-6\right)-6}{x-6}=0$$

Do the multiplications in x\left(x-6\right)-6.

$$\frac{x^{2}-6x-6}{x-6}=0$$

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by x-6.

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

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

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-6\right)}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$x=\frac{-\left(-6\right)±\sqrt{36+24}}{2}$$

Add 36 to 24.

$$x=\frac{-\left(-6\right)±\sqrt{60}}{2}$$

Take the square root of 60.

$$x=\frac{-\left(-6\right)±2\sqrt{15}}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{15}}{2}$$

Now solve the equation x=\frac{6±2\sqrt{15}}{2} when ± is plus. Add 6 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}+6}{2}$$

Divide 6+2\sqrt{15} by 2.

$$x=\sqrt{15}+3$$

Now solve the equation x=\frac{6±2\sqrt{15}}{2} when ± is minus. Subtract 2\sqrt{15} from 6.

$$x=\frac{6-2\sqrt{15}}{2}$$

Divide 6-2\sqrt{15} by 2.

$$x=3-\sqrt{15}$$

The equation is now solved.

$$x=\sqrt{15}+3 x=3-\sqrt{15}$$

Steps for Completing the Square

Subtract \frac{6}{x-6} from both sides.

$$x-\frac{6}{x-6}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-6}{x-6}.

$$\frac{x\left(x-6\right)}{x-6}-\frac{6}{x-6}=0$$

Since \frac{x\left(x-6\right)}{x-6} and \frac{6}{x-6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x-6\right)-6}{x-6}=0$$

Do the multiplications in x\left(x-6\right)-6.

$$\frac{x^{2}-6x-6}{x-6}=0$$

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by x-6.

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

Add 6 to both sides. Anything plus zero gives itself.

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

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=6+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=6+9$$

Add 6 to 9.

$$x^{2}-6x+9=15$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=15$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{15}$$

Simplify.

$$x-3=\sqrt{15} x-3=-\sqrt{15}$$

Add 3 to both sides of the equation.

$$x=\sqrt{15}+3 x=3-\sqrt{15}$$