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

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

Solve for x (complex solution)

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

Steps Using the Quadratic Formula

Substitute t for x^{3}.

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

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, 1 for b, and -6 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-1±5}{2}$$

Solve the equation t=\frac{-1±5}{2} when ± is plus and when ± is minus.

$$t=2 t=-3$$

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

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

Solve for x

$$x=-\sqrt[3]{3}\approx -1.44224957$$

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

Steps Using the Quadratic Formula

Substitute t for x^{3}.

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

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, 1 for b, and -6 for c in the quadratic formula.

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

Do the calculations.

$$t=\frac{-1±5}{2}$$

Solve the equation t=\frac{-1±5}{2} when ± is plus and when ± is minus.

$$t=2 t=-3$$

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

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