Solve x^6-19x^3-216=0

$$x ^ { 6 } - 19 x ^ { 3 } - 216 = 0$$

Solve for x (complex solution)

$$x\in 3,\frac{-3+3\sqrt{3}i}{2},\frac{-3\sqrt{3}i-3}{2},1+\sqrt{3}i,-2,-\sqrt{3}i+1$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$t^{2}-19t-216=0$$

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, -19 for b, and -216 for c in the quadratic formula.

$$t=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 1\left(-216\right)}}{2}$$

Do the calculations.

$$t=\frac{19±35}{2}$$

Solve the equation t=\frac{19±35}{2} when ± is plus and when ± is minus.

$$t=27 t=-8$$

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

$$x=\frac{-3+3\sqrt{3}i}{2} x=\frac{-3\sqrt{3}i-3}{2} x=3 x=1+\sqrt{3}i x=-\sqrt{3}i+1 x=-2$$

Solve for x

$$x=-2$$

$$x=3$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$t^{2}-19t-216=0$$

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, -19 for b, and -216 for c in the quadratic formula.

$$t=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 1\left(-216\right)}}{2}$$

Do the calculations.

$$t=\frac{19±35}{2}$$

Solve the equation t=\frac{19±35}{2} when ± is plus and when ± is minus.

$$t=27 t=-8$$

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

$$x=3 x=-2$$