Solve x^5+1/x^5=123

$${ x }^{ 5 } + \frac{ 1 }{ { x }^{ 5 } } =123$$

Solve for x

$$x=\frac{2^{\frac{4}{5}}\sqrt[5]{123-55\sqrt{5}}}{2}\approx 0.381966011$$

$$x = \frac{2 ^ {\frac{4}{5}} \sqrt[5]{55 \sqrt{5} + 123}}{2} \approx 2.618033989$$

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^{5}.

$$x^{5}x^{5}+1=123x^{5}$$

To multiply powers of the same base, add their exponents. Add 5 and 5 to get 10.

$$x^{10}+1=123x^{5}$$

Subtract 123x^{5} from both sides.

$$x^{10}+1-123x^{5}=0$$

Substitute t for x^{5}.

$$t^{2}-123t+1=0$$

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, -123 for b, and 1 for c in the quadratic formula.

$$t=\frac{-\left(-123\right)±\sqrt{\left(-123\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{123±55\sqrt{5}}{2}$$

Solve the equation t=\frac{123±55\sqrt{5}}{2} when ± is plus and when ± is minus.

$$t=\frac{55\sqrt{5}+123}{2} t=\frac{123-55\sqrt{5}}{2}$$

Since x=t^{5}, the solutions are obtained by evaluating x=\sqrt[5]{t} for each t.

$$x=\sqrt[5]{\frac{55\sqrt{5}+123}{2}} x=\sqrt[5]{\frac{123-55\sqrt{5}}{2}}$$