Solve x=1/x-3

$$x= \frac{ 1 }{ x-3 }$$

Solve for x

$$x = \frac{\sqrt{13} + 3}{2} \approx 3.302775638$$

$$x=\frac{3-\sqrt{13}}{2}\approx -0.302775638$$

Steps Using the Quadratic Formula

Subtract \frac{1}{x-3} from both sides.

$$x-\frac{1}{x-3}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-3}{x-3}.

$$\frac{x\left(x-3\right)}{x-3}-\frac{1}{x-3}=0$$

Since \frac{x\left(x-3\right)}{x-3} and \frac{1}{x-3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x-3\right)-1}{x-3}=0$$

Do the multiplications in x\left(x-3\right)-1.

$$\frac{x^{2}-3x-1}{x-3}=0$$

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by x-3.

$$x^{2}-3x-1=0$$

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

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

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

Add 9 to 4.

$$x=\frac{-\left(-3\right)±\sqrt{13}}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{13}}{2}$$

Now solve the equation x=\frac{3±\sqrt{13}}{2} when ± is plus. Add 3 to \sqrt{13}.

$$x=\frac{\sqrt{13}+3}{2}$$

Now solve the equation x=\frac{3±\sqrt{13}}{2} when ± is minus. Subtract \sqrt{13} from 3.

$$x=\frac{3-\sqrt{13}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{13}+3}{2} x=\frac{3-\sqrt{13}}{2}$$

Steps for Completing the Square

Subtract \frac{1}{x-3} from both sides.

$$x-\frac{1}{x-3}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-3}{x-3}.

$$\frac{x\left(x-3\right)}{x-3}-\frac{1}{x-3}=0$$

Since \frac{x\left(x-3\right)}{x-3} and \frac{1}{x-3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x-3\right)-1}{x-3}=0$$

Do the multiplications in x\left(x-3\right)-1.

$$\frac{x^{2}-3x-1}{x-3}=0$$

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by x-3.

$$x^{2}-3x-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

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

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.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=1+\left(-\frac{3}{2}\right)^{2}$$

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

$$x^{2}-3x+\frac{9}{4}=1+\frac{9}{4}$$

Add 1 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{13}{4}$$

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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{13}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{13}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{13}}{2} x-\frac{3}{2}=-\frac{\sqrt{13}}{2}$$

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

$$x=\frac{\sqrt{13}+3}{2} x=\frac{3-\sqrt{13}}{2}$$