Solve 27x^6+8=35x^3

$$27 x ^ { 6 } + 8 = 35 x ^ { 3 }$$

Solve for x (complex solution)

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

Steps Using the Quadratic Formula

Subtract 35x^{3} from both sides.

$$27x^{6}+8-35x^{3}=0$$

Substitute t for x^{3}.

$$27t^{2}-35t+8=0$$

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, -35 for b, and 8 for c in the quadratic formula.

$$t=\frac{-\left(-35\right)±\sqrt{\left(-35\right)^{2}-4\times 27\times 8}}{2\times 27}$$

Do the calculations.

$$t=\frac{35±19}{54}$$

Solve the equation t=\frac{35±19}{54} when ± is plus and when ± is minus.

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

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

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

Solve for x

$$x=1$$

$$x=\frac{2}{3}\approx 0.666666667$$

Steps Using the Quadratic Formula

Subtract 35x^{3} from both sides.

$$27x^{6}+8-35x^{3}=0$$

Substitute t for x^{3}.

$$27t^{2}-35t+8=0$$

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, -35 for b, and 8 for c in the quadratic formula.

$$t=\frac{-\left(-35\right)±\sqrt{\left(-35\right)^{2}-4\times 27\times 8}}{2\times 27}$$

Do the calculations.

$$t=\frac{35±19}{54}$$

Solve the equation t=\frac{35±19}{54} when ± is plus and when ± is minus.

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

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

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