Solve 8/x^6-217/x^3+27=0

$$\frac{ 8 }{ { x }^{ 6 } } - \frac{ 217 }{ { x }^{ 3 } } +27=0$$

Solve for x (complex solution)

$$x\in \frac{-\sqrt{3}i-1}{6},\frac{-1+\sqrt{3}i}{6},\frac{1}{3},-1+\sqrt{3}i,-\sqrt{3}i-1,2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{6}, the least common multiple of x^{6},x^{3}.

$$8-x^{3}\times 217+x^{6}\times 27=0$$

Multiply -1 and 217 to get -217.

$$8-217x^{3}+x^{6}\times 27=0$$

Substitute t for x^{3}.

$$27t^{2}-217t+8=0$$

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 27 for a, -217 for b, and 8 for c in the quadratic formula.

$$t=\frac{-\left(-217\right)±\sqrt{\left(-217\right)^{2}-4\times 27\times 8}}{2\times 27}$$

Do the calculations.

$$t=\frac{217±215}{54}$$

Solve the equation t=\frac{217±215}{54} when ± is plus and when ± is minus.

$$t=8 t=\frac{1}{27}$$

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

$$x=-1+\sqrt{3}i x=-\sqrt{3}i-1 x=2 x=\frac{-1+\sqrt{3}i}{6} x=\frac{-\sqrt{3}i-1}{6} x=\frac{1}{3}$$

Solve for x

$$x=2$$

$$x=\frac{1}{3}\approx 0.333333333$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x^{6}, the least common multiple of x^{6},x^{3}.

$$8-x^{3}\times 217+x^{6}\times 27=0$$

Multiply -1 and 217 to get -217.

$$8-217x^{3}+x^{6}\times 27=0$$

Substitute t for x^{3}.

$$27t^{2}-217t+8=0$$

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 27 for a, -217 for b, and 8 for c in the quadratic formula.

$$t=\frac{-\left(-217\right)±\sqrt{\left(-217\right)^{2}-4\times 27\times 8}}{2\times 27}$$

Do the calculations.

$$t=\frac{217±215}{54}$$

Solve the equation t=\frac{217±215}{54} when ± is plus and when ± is minus.

$$t=8 t=\frac{1}{27}$$

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

$$x=2 x=\frac{1}{3}$$