Solve {left(x-1/xright)}^2=20

$${ \left(x- \frac{ 1 }{ x } \right) }^{ 2 } =20$$

Solve for x

$$x=\sqrt{5}-\sqrt{6}\approx -0.213421765$$

$$x=\sqrt{5}+\sqrt{6}\approx 4.68555772$$

$$x=\sqrt{6}-\sqrt{5}\approx 0.213421765$$

$$x=-\left(\sqrt{5}+\sqrt{6}\right)\approx -4.68555772$$

Steps Using the Quadratic Formula

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

$$\left(\frac{xx}{x}-\frac{1}{x}\right)^{2}=20$$

Since \frac{xx}{x} and \frac{1}{x} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{xx-1}{x}\right)^{2}=20$$

Do the multiplications in xx-1.

$$\left(\frac{x^{2}-1}{x}\right)^{2}=20$$

To raise \frac{x^{2}-1}{x} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(x^{2}-1\right)^{2}}{x^{2}}=20$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-1\right)^{2}.

$$\frac{\left(x^{2}\right)^{2}-2x^{2}+1}{x^{2}}=20$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{x^{4}-2x^{2}+1}{x^{2}}=20$$

Subtract 20 from both sides.

$$\frac{x^{4}-2x^{2}+1}{x^{2}}-20=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 20 times \frac{x^{2}}{x^{2}}.

$$\frac{x^{4}-2x^{2}+1}{x^{2}}-\frac{20x^{2}}{x^{2}}=0$$

Since \frac{x^{4}-2x^{2}+1}{x^{2}} and \frac{20x^{2}}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{4}-2x^{2}+1-20x^{2}}{x^{2}}=0$$

Combine like terms in x^{4}-2x^{2}+1-20x^{2}.

$$\frac{x^{4}-22x^{2}+1}{x^{2}}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{2}.

$$x^{4}-22x^{2}+1=0$$

Substitute t for x^{2}.

$$t^{2}-22t+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, -22 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{22±4\sqrt{30}}{2}$$

Solve the equation t=\frac{22±4\sqrt{30}}{2} when ± is plus and when ± is minus.

$$t=2\sqrt{30}+11 t=11-2\sqrt{30}$$

Since x=t^{2}, the solutions are obtained by evaluating x=±\sqrt{t} for each t.

$$x=\sqrt{5}+\sqrt{6} x=-\left(\sqrt{5}+\sqrt{6}\right) x=-\left(\sqrt{5}-\sqrt{6}\right) x=\sqrt{5}-\sqrt{6}$$