Solve x^10+5x^5-1=0

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

Solve for x

$$x=\frac{2^{\frac{4}{5}}\sqrt[5]{\sqrt{29}-5}}{2}\approx 0.719321963$$

$$x = \frac{2 ^ {\frac{4}{5}} \sqrt[5]{-\sqrt{29} - 5}}{2} \approx 1.390198064$$

Steps Using the Quadratic Formula

Substitute t for x^{5}.

$$t^{2}+5t-1=0$$

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

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

Do the calculations.

$$t=\frac{-5±\sqrt{29}}{2}$$

Solve the equation t=\frac{-5±\sqrt{29}}{2} when ± is plus and when ± is minus.

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

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

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