Solve x-1/x=9

$$x - \frac { 1 } { x } = 9$$

Solve for x

$$x = \frac{\sqrt{85} + 9}{2} \approx 9.109772229$$

$$x=\frac{9-\sqrt{85}}{2}\approx -0.109772229$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$xx-1=9x$$

Multiply x and x to get x^{2}.

$$x^{2}-1=9x$$

Subtract 9x from both sides.

$$x^{2}-1-9x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-9x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$x=\frac{-\left(-9\right)±\sqrt{81+4}}{2}$$

Add 81 to 4.

$$x=\frac{-\left(-9\right)±\sqrt{85}}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{85}}{2}$$

Now solve the equation x=\frac{9±\sqrt{85}}{2} when ± is plus. Add 9 to \sqrt{85}.

$$x=\frac{\sqrt{85}+9}{2}$$

Now solve the equation x=\frac{9±\sqrt{85}}{2} when ± is minus. Subtract \sqrt{85} from 9.

$$x=\frac{9-\sqrt{85}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{85}+9}{2} x=\frac{9-\sqrt{85}}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$xx-1=9x$$

Multiply x and x to get x^{2}.

$$x^{2}-1=9x$$

Subtract 9x from both sides.

$$x^{2}-1-9x=0$$

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

$$x^{2}-9x=1$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=1+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=1+\frac{81}{4}$$

Add 1 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{85}{4}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{85}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{85}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{85}}{2} x-\frac{9}{2}=-\frac{\sqrt{85}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{\sqrt{85}+9}{2} x=\frac{9-\sqrt{85}}{2}$$