Solve 6x^6+x^3-5=0

$$6 x ^ { 6 } + x ^ { 3 } - 5 = 0$$

Solve for x (complex solution)

$$x\in \frac{\sqrt[3]{5}\times 6^{\frac{2}{3}}}{6},\frac{\sqrt[3]{5}\times 6^{\frac{2}{3}}e^{\frac{4\pi i}{3}}}{6},\frac{\sqrt[3]{5}\times 6^{\frac{2}{3}}e^{\frac{2\pi i}{3}}}{6},-1,\frac{-\sqrt{3}i+1}{2},\frac{1+\sqrt{3}i}{2}$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$6t^{2}+t-5=0$$

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, 1 for b, and -5 for c in the quadratic formula.

$$t=\frac{-1±\sqrt{1^{2}-4\times 6\left(-5\right)}}{2\times 6}$$

Do the calculations.

$$t=\frac{-1±11}{12}$$

Solve the equation t=\frac{-1±11}{12} when ± is plus and when ± is minus.

$$t=\frac{5}{6} t=-1$$

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

$$x=-\frac{\sqrt[3]{5}\times 6^{\frac{2}{3}}e^{\frac{\pi i}{3}}}{6} x=\frac{\sqrt[3]{5}\times 6^{\frac{2}{3}}ie^{\frac{\pi i}{6}}}{6} x=\frac{\sqrt[3]{5}\times 6^{\frac{2}{3}}}{6} x=-1 x=\frac{1+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i+1}{2}$$

Solve for x

$$x=-1$$

$$x=\frac{\sqrt[3]{5}\times 6^{\frac{2}{3}}}{6}\approx 0.941036029$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$6t^{2}+t-5=0$$

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, 1 for b, and -5 for c in the quadratic formula.

$$t=\frac{-1±\sqrt{1^{2}-4\times 6\left(-5\right)}}{2\times 6}$$

Do the calculations.

$$t=\frac{-1±11}{12}$$

Solve the equation t=\frac{-1±11}{12} when ± is plus and when ± is minus.

$$t=\frac{5}{6} t=-1$$

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

$$x=\frac{\sqrt[3]{5}\times 6^{\frac{2}{3}}}{6} x=-1$$