Solve 4=1/6x^6+x^3+2

$$4 = \frac { 1 } { 6 } x ^ { 6 } + x ^ { 3 } + 2$$

Solve for x (complex solution)

$$x\in \sqrt[3]{\sqrt{21}+3}e^{\frac{\pi i}{3}},\sqrt[3]{\sqrt{21}+3}e^{\frac{5\pi i}{3}},-\sqrt[3]{\sqrt{21}+3},\sqrt[3]{\sqrt{21}-3}e^{\frac{4\pi i}{3}},\sqrt[3]{\sqrt{21}-3},\sqrt[3]{\sqrt{21}-3}e^{\frac{2\pi i}{3}}$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{6}x^{6}+x^{3}+2=4$$

Subtract 4 from both sides.

$$\frac{1}{6}x^{6}+x^{3}+2-4=0$$

Subtract 4 from 2 to get -2.

$$\frac{1}{6}x^{6}+x^{3}-2=0$$

Substitute t for x^{3}.

$$\frac{1}{6}t^{2}+t-2=0$$

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 \frac{1}{6} for a, 1 for b, and -2 for c in the quadratic formula.

$$t=\frac{-1±\sqrt{1^{2}-4\times \frac{1}{6}\left(-2\right)}}{\frac{1}{6}\times 2}$$

Do the calculations.

$$t=\frac{-1±\frac{1}{3}\sqrt{21}}{\frac{1}{3}}$$

Solve the equation t=\frac{-1±\frac{1}{3}\sqrt{21}}{\frac{1}{3}} when ± is plus and when ± is minus.

$$t=\sqrt{21}-3 t=-\sqrt{21}-3$$

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

$$x=-\sqrt[3]{\sqrt{21}-3}e^{\frac{\pi i}{3}} x=\sqrt[3]{\sqrt{21}-3}ie^{\frac{\pi i}{6}} x=\sqrt[3]{\sqrt{21}-3} x=-\sqrt[3]{\sqrt{21}+3}ie^{\frac{\pi i}{6}} x=-\sqrt[3]{\sqrt{21}+3} x=\sqrt[3]{\sqrt{21}+3}e^{\frac{\pi i}{3}}$$

Solve for x

$$x=\sqrt[3]{\sqrt{21}-3}\approx 1.165345841$$

$$x=-\sqrt[3]{\sqrt{21}+3}\approx -1.964591458$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{6}x^{6}+x^{3}+2=4$$

Subtract 4 from both sides.

$$\frac{1}{6}x^{6}+x^{3}+2-4=0$$

Subtract 4 from 2 to get -2.

$$\frac{1}{6}x^{6}+x^{3}-2=0$$

Substitute t for x^{3}.

$$\frac{1}{6}t^{2}+t-2=0$$

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 \frac{1}{6} for a, 1 for b, and -2 for c in the quadratic formula.

$$t=\frac{-1±\sqrt{1^{2}-4\times \frac{1}{6}\left(-2\right)}}{\frac{1}{6}\times 2}$$

Do the calculations.

$$t=\frac{-1±\frac{1}{3}\sqrt{21}}{\frac{1}{3}}$$

Solve the equation t=\frac{-1±\frac{1}{3}\sqrt{21}}{\frac{1}{3}} when ± is plus and when ± is minus.

$$t=\sqrt{21}-3 t=-\sqrt{21}-3$$

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

$$x=\sqrt[3]{\sqrt{21}-3} x=-\sqrt[3]{\sqrt{21}+3}$$