Solve 6x^6-3x^3-28=0

$$6 x ^ { 6 } - 3 x ^ { 3 } - 28 = 0$$

Solve for x (complex solution)

$$x\in \frac{12^{\frac{2}{3}}\sqrt[3]{\sqrt{681}-3}e^{\frac{\pi i}{3}}}{12},\frac{12^{\frac{2}{3}}\sqrt[3]{\sqrt{681}-3}e^{\frac{5\pi i}{3}}}{12},-\frac{12^{\frac{2}{3}}\sqrt[3]{\sqrt{681}-3}}{12},\frac{12^{\frac{2}{3}}\sqrt[3]{\sqrt{681}+3}e^{\frac{4\pi i}{3}}}{12},\frac{12^{\frac{2}{3}}\sqrt[3]{\sqrt{681}+3}}{12},\frac{12^{\frac{2}{3}}\sqrt[3]{\sqrt{681}+3}e^{\frac{2\pi i}{3}}}{12}$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$6t^{2}-3t-28=0$$

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 6 for a, -3 for b, and -28 for c in the quadratic formula.

$$t=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 6\left(-28\right)}}{2\times 6}$$

Do the calculations.

$$t=\frac{3±\sqrt{681}}{12}$$

Solve the equation t=\frac{3±\sqrt{681}}{12} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{681}}{12}+\frac{1}{4} t=-\frac{\sqrt{681}}{12}+\frac{1}{4}$$

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

$$x=-\sqrt[3]{\frac{\sqrt{681}}{12}+\frac{1}{4}}e^{\frac{\pi i}{3}} x=\sqrt[3]{\frac{\sqrt{681}}{12}+\frac{1}{4}}ie^{\frac{\pi i}{6}} x=\sqrt[3]{\frac{\sqrt{681}}{12}+\frac{1}{4}} x=-\sqrt[3]{-\left(-\frac{\sqrt{681}}{12}+\frac{1}{4}\right)}ie^{\frac{\pi i}{6}} x=-\sqrt[3]{-\left(-\frac{\sqrt{681}}{12}+\frac{1}{4}\right)} x=\sqrt[3]{-\left(-\frac{\sqrt{681}}{12}+\frac{1}{4}\right)}e^{\frac{\pi i}{3}}$$

Solve for x

$$x = \frac{12 ^ {\frac{2}{3}} \sqrt[3]{\sqrt{681} + 3}}{12} \approx 1.343436772$$

$$x = \frac{12 ^ {\frac{2}{3}} \sqrt[3]{3 - \sqrt{681}}}{12} \approx 1.243898743$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$6t^{2}-3t-28=0$$

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 6 for a, -3 for b, and -28 for c in the quadratic formula.

$$t=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 6\left(-28\right)}}{2\times 6}$$

Do the calculations.

$$t=\frac{3±\sqrt{681}}{12}$$

Solve the equation t=\frac{3±\sqrt{681}}{12} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{681}}{12}+\frac{1}{4} t=-\frac{\sqrt{681}}{12}+\frac{1}{4}$$

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

$$x=\sqrt[3]{\frac{\sqrt{681}}{12}+\frac{1}{4}} x=\sqrt[3]{-\frac{\sqrt{681}}{12}+\frac{1}{4}}$$