Solve x^1-x^-1=5

$$x ^ { 1 } - x ^ { - 1 } = 5$$

Solve for x

$$x=\frac{5-\sqrt{29}}{2}\approx -0.192582404$$

$$x = \frac{\sqrt{29} + 5}{2} \approx 5.192582404$$

Steps Using the Quadratic Formula

Calculate x to the power of 1 and get x.

$$x-x^{-1}=5$$

Subtract 5 from both sides.

$$x-x^{-1}-5=0$$

Reorder the terms.

$$x-5-\frac{1}{x}=0$$

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

$$xx+x\left(-5\right)-1=0$$

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

$$x^{2}+x\left(-5\right)-1=0$$

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.

$$x^{2}-5x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)}}{2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$x=\frac{-\left(-5\right)±\sqrt{25+4}}{2}$$

Add 25 to 4.

$$x=\frac{-\left(-5\right)±\sqrt{29}}{2}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{29}}{2}$$

Now solve the equation x=\frac{5±\sqrt{29}}{2} when ± is plus. Add 5 to \sqrt{29}.

$$x=\frac{\sqrt{29}+5}{2}$$

Now solve the equation x=\frac{5±\sqrt{29}}{2} when ± is minus. Subtract \sqrt{29} from 5.

$$x=\frac{5-\sqrt{29}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{29}+5}{2} x=\frac{5-\sqrt{29}}{2}$$

Steps for Completing the Square

Calculate x to the power of 1 and get x.

$$x-x^{-1}=5$$

Reorder the terms.

$$x-\frac{1}{x}=5$$

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

$$xx-1=5x$$

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

$$x^{2}-1=5x$$

Subtract 5x from both sides.

$$x^{2}-1-5x=0$$

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

$$x^{2}-5x=1$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=1+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=1+\frac{25}{4}$$

Add 1 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{29}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{29}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{29}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{29}}{2} x-\frac{5}{2}=-\frac{\sqrt{29}}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{\sqrt{29}+5}{2} x=\frac{5-\sqrt{29}}{2}$$