Solve 2x^6-x^3-1=0

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

Solve for x (complex solution)

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

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$2t^{2}-t-1=0$$

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 2 for a, -1 for b, and -1 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{1±3}{4}$$

Solve the equation t=\frac{1±3}{4} when ± is plus and when ± is minus.

$$t=1 t=-\frac{1}{2}$$

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

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

Solve for x

$$x=-\frac{2^{\frac{2}{3}}}{2}\approx -0.793700526$$

$$x=1$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$2t^{2}-t-1=0$$

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 2 for a, -1 for b, and -1 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{1±3}{4}$$

Solve the equation t=\frac{1±3}{4} when ± is plus and when ± is minus.

$$t=1 t=-\frac{1}{2}$$

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

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