Solve x^6-9x^3+8=0

$${ x }^{ 6 } -9 { x }^{ 3 } +8=0$$

Solve for x (complex solution)

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

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$t^{2}-9t+8=0$$

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 1 for a, -9 for b, and 8 for c in the quadratic formula.

$$t=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 1\times 8}}{2}$$

Do the calculations.

$$t=\frac{9±7}{2}$$

Solve the equation t=\frac{9±7}{2} when ± is plus and when ± is minus.

$$t=8 t=1$$

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

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

Solve for x

$$x=1$$

$$x=2$$

Steps Using the Quadratic Formula

Substitute t for x^{3}.

$$t^{2}-9t+8=0$$

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 1 for a, -9 for b, and 8 for c in the quadratic formula.

$$t=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 1\times 8}}{2}$$

Do the calculations.

$$t=\frac{9±7}{2}$$

Solve the equation t=\frac{9±7}{2} when ± is plus and when ± is minus.

$$t=8 t=1$$

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

$$x=2 x=1$$