Solve x^10-33x^5+32=0

$$x ^ { 10 } - 33 x ^ { 5 } + 32 = 0$$

Solve for x (complex solution)

$$x\in 2,2e^{\frac{8\pi i}{5}},2e^{\frac{2\pi i}{5}},2e^{\frac{4\pi i}{5}},2e^{\frac{6\pi i}{5}},e^{\frac{8\pi i}{5}},1,e^{\frac{2\pi i}{5}},e^{\frac{4\pi i}{5}},e^{\frac{6\pi i}{5}}$$

Steps Using the Quadratic Formula

Substitute t for x^{5}.

$$t^{2}-33t+32=0$$

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 32 for c in the quadratic formula.

$$t=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 1\times 32}}{2}$$

Do the calculations.

$$t=\frac{33±31}{2}$$

Solve the equation t=\frac{33±31}{2} when ± is plus and when ± is minus.

$$t=32 t=1$$

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

$$x=-2ie^{\frac{\pi i}{10}} x=-2e^{\frac{\pi i}{5}} x=2ie^{\frac{3\pi i}{10}} x=2e^{\frac{2\pi i}{5}} x=2 x=-ie^{\frac{\pi i}{10}} x=-e^{\frac{\pi i}{5}} x=ie^{\frac{3\pi i}{10}} x=e^{\frac{2\pi i}{5}} x=1$$

Solve for x

$$x=1$$

$$x=2$$

Steps Using the Quadratic Formula

Substitute t for x^{5}.

$$t^{2}-33t+32=0$$

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 32 for c in the quadratic formula.

$$t=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 1\times 32}}{2}$$

Do the calculations.

$$t=\frac{33±31}{2}$$

Solve the equation t=\frac{33±31}{2} when ± is plus and when ± is minus.

$$t=32 t=1$$

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

$$x=2 x=1$$