Solve x^10-13x^5+36=0

$$x ^ { 10 } - 13 x ^ { 5 } + 36 = 0$$

Solve for x (complex solution)

$$x\in 3^{\frac{2}{5}},3^{\frac{2}{5}}e^{\frac{8\pi i}{5}},3^{\frac{2}{5}}e^{\frac{2\pi i}{5}},3^{\frac{2}{5}}e^{\frac{4\pi i}{5}},3^{\frac{2}{5}}e^{\frac{6\pi i}{5}},\sqrt[5]{4}e^{\frac{8\pi i}{5}},\sqrt[5]{4},\sqrt[5]{4}e^{\frac{2\pi i}{5}},\sqrt[5]{4}e^{\frac{4\pi i}{5}},\sqrt[5]{4}e^{\frac{6\pi i}{5}}$$

Steps Using the Quadratic Formula

Substitute t for x^{5}.

$$t^{2}-13t+36=0$$

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, -13 for b, and 36 for c in the quadratic formula.

$$t=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 1\times 36}}{2}$$

Do the calculations.

$$t=\frac{13±5}{2}$$

Solve the equation t=\frac{13±5}{2} when ± is plus and when ± is minus.

$$t=9 t=4$$

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

$$x=-\sqrt[5]{9}ie^{\frac{\pi i}{10}} x=-\sqrt[5]{9}e^{\frac{\pi i}{5}} x=\sqrt[5]{9}ie^{\frac{3\pi i}{10}} x=\sqrt[5]{9}e^{\frac{2\pi i}{5}} x=\sqrt[5]{9} x=-\sqrt[5]{4}ie^{\frac{\pi i}{10}} x=-\sqrt[5]{4}e^{\frac{\pi i}{5}} x=\sqrt[5]{4}ie^{\frac{3\pi i}{10}} x=\sqrt[5]{4}e^{\frac{2\pi i}{5}} x=\sqrt[5]{4}$$

Solve for x

$$x=\sqrt[5]{4}\approx 1.319507911$$

$$x=3^{\frac{2}{5}}\approx 1.551845574$$

Steps Using the Quadratic Formula

Substitute t for x^{5}.

$$t^{2}-13t+36=0$$

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, -13 for b, and 36 for c in the quadratic formula.

$$t=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 1\times 36}}{2}$$

Do the calculations.

$$t=\frac{13±5}{2}$$

Solve the equation t=\frac{13±5}{2} when ± is plus and when ± is minus.

$$t=9 t=4$$

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

$$x=\sqrt[5]{9} x=\sqrt[5]{4}$$