Solve x^10+4x^5+3=0

$$x ^ { 10 } + 4 x ^ { 5 } + 3 = 0$$

Solve for x (complex solution)

$$x\in e^{\frac{\pi i}{5}},e^{\frac{9\pi i}{5}},e^{\frac{3\pi i}{5}},-1,e^{\frac{7\pi i}{5}},\sqrt[5]{3}e^{\frac{9\pi i}{5}},\sqrt[5]{3}e^{\frac{\pi i}{5}},\sqrt[5]{3}e^{\frac{3\pi i}{5}},-\sqrt[5]{3},\sqrt[5]{3}e^{\frac{7\pi i}{5}}$$

Steps Using the Quadratic Formula

Substitute t for x^{5}.

$$t^{2}+4t+3=0$$

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, 4 for b, and 3 for c in the quadratic formula.

$$t=\frac{-4±\sqrt{4^{2}-4\times 1\times 3}}{2}$$

Do the calculations.

$$t=\frac{-4±2}{2}$$

Solve the equation t=\frac{-4±2}{2} when ± is plus and when ± is minus.

$$t=-1 t=-3$$

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

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

Solve for x

$$x=-\sqrt[5]{3}\approx -1.24573094$$

$$x=-1$$

Steps Using the Quadratic Formula

Substitute t for x^{5}.

$$t^{2}+4t+3=0$$

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, 4 for b, and 3 for c in the quadratic formula.

$$t=\frac{-4±\sqrt{4^{2}-4\times 1\times 3}}{2}$$

Do the calculations.

$$t=\frac{-4±2}{2}$$

Solve the equation t=\frac{-4±2}{2} when ± is plus and when ± is minus.

$$t=-1 t=-3$$

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

$$x=-1 x=-\sqrt[5]{3}$$