Solve x^6+3x^3+2=0

$$x ^ { 6 } + 3 x ^ { 3 } + 2 = 0$$

Solve for x (complex solution)

$$x\in \frac{-\sqrt{3}i+1}{2},-1,\frac{1+\sqrt{3}i}{2},\sqrt[3]{2}e^{\frac{5\pi i}{3}},\sqrt[3]{2}e^{\frac{\pi i}{3}},-\sqrt[3]{2}$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

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

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, 3 for b, and 2 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-3±1}{2}$$

Solve the equation t=\frac{-3±1}{2} when ± is plus and when ± is minus.

$$t=-1 t=-2$$

Since x=t^{3}, the solutions are obtained by solving the equation for each t.

$$x=-1 x=\frac{1+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i+1}{2} x=-\sqrt[3]{2}ie^{\frac{\pi i}{6}} x=-\sqrt[3]{2} x=\sqrt[3]{2}e^{\frac{\pi i}{3}}$$

Solve for x

$$x=-\sqrt[3]{2}\approx -1.25992105$$

$$x=-1$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

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

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, 3 for b, and 2 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-3±1}{2}$$

Solve the equation t=\frac{-3±1}{2} when ± is plus and when ± is minus.

$$t=-1 t=-2$$

Since x=t^{3}, the solutions are obtained by evaluating x=\sqrt[3]{t} for each t.

$$x=-1 x=-\sqrt[3]{2}$$