Solve x^10-33x^5+52=0

$$x ^ { 10 } - 33 x ^ { 5 } + 52 = 0$$

Solve for x (complex solution)

$$x\in \frac{2^{\frac{4}{5}}\sqrt[5]{33-\sqrt{881}}}{2},\frac{2^{\frac{4}{5}}e^{\frac{8\pi i}{5}}\sqrt[5]{33-\sqrt{881}}}{2},\frac{2^{\frac{4}{5}}e^{\frac{2\pi i}{5}}\sqrt[5]{33-\sqrt{881}}}{2},\frac{2^{\frac{4}{5}}e^{\frac{4\pi i}{5}}\sqrt[5]{33-\sqrt{881}}}{2},\frac{2^{\frac{4}{5}}e^{\frac{6\pi i}{5}}\sqrt[5]{33-\sqrt{881}}}{2},\frac{2^{\frac{4}{5}}e^{\frac{8\pi i}{5}}\sqrt[5]{\sqrt{881}+33}}{2},\frac{2^{\frac{4}{5}}\sqrt[5]{\sqrt{881}+33}}{2},\frac{2^{\frac{4}{5}}e^{\frac{2\pi i}{5}}\sqrt[5]{\sqrt{881}+33}}{2},\frac{2^{\frac{4}{5}}e^{\frac{4\pi i}{5}}\sqrt[5]{\sqrt{881}+33}}{2},\frac{2^{\frac{4}{5}}e^{\frac{6\pi i}{5}}\sqrt[5]{\sqrt{881}+33}}{2}$$

Steps Using the Quadratic Formula

Substitute t for x^{5}.

$$t^{2}-33t+52=0$$

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, -33 for b, and 52 for c in the quadratic formula.

$$t=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 1\times 52}}{2}$$

Do the calculations.

$$t=\frac{33±\sqrt{881}}{2}$$

Solve the equation t=\frac{33±\sqrt{881}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{881}+33}{2} t=\frac{33-\sqrt{881}}{2}$$

Since x=t^{5}, the solutions are obtained by solving the equation for each t.

$$x=-ie^{\frac{\pi i}{10}}\sqrt[5]{\frac{\sqrt{881}+33}{2}} x=-e^{\frac{\pi i}{5}}\sqrt[5]{\frac{\sqrt{881}+33}{2}} x=ie^{\frac{3\pi i}{10}}\sqrt[5]{\frac{\sqrt{881}+33}{2}} x=e^{\frac{2\pi i}{5}}\sqrt[5]{\frac{\sqrt{881}+33}{2}} x=\sqrt[5]{\frac{\sqrt{881}+33}{2}} x=-ie^{\frac{\pi i}{10}}\sqrt[5]{\frac{33-\sqrt{881}}{2}} x=-e^{\frac{\pi i}{5}}\sqrt[5]{\frac{33-\sqrt{881}}{2}} x=ie^{\frac{3\pi i}{10}}\sqrt[5]{\frac{33-\sqrt{881}}{2}} x=e^{\frac{2\pi i}{5}}\sqrt[5]{\frac{33-\sqrt{881}}{2}} x=\sqrt[5]{\frac{33-\sqrt{881}}{2}}$$

Solve for x

$$x = \frac{2 ^ {\frac{4}{5}} \sqrt[5]{\sqrt{881} + 33}}{2} \approx 1.991691532$$

$$x = \frac{2 ^ {\frac{4}{5}} \sqrt[5]{33 - \sqrt{881}}}{2} \approx 1.106569236$$

Steps Using the Quadratic Formula

Substitute t for x^{5}.

$$t^{2}-33t+52=0$$

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, -33 for b, and 52 for c in the quadratic formula.

$$t=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 1\times 52}}{2}$$

Do the calculations.

$$t=\frac{33±\sqrt{881}}{2}$$

Solve the equation t=\frac{33±\sqrt{881}}{2} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{881}+33}{2} t=\frac{33-\sqrt{881}}{2}$$

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

$$x=\sqrt[5]{\frac{\sqrt{881}+33}{2}} x=\sqrt[5]{\frac{33-\sqrt{881}}{2}}$$