Solve x^6=((6x^3)-5^3)

$${ x }^{ 6 } =((6 { x }^{ 3 } )- { 5 }^{ 3 } )$$

Solve for x (complex solution)

$$x\in \sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}},\sqrt{5}e^{-\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}},\sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}},\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}},\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}},\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}}$$

Steps Using the Quadratic Formula

Calculate 5 to the power of 3 and get 125.

$$x^{6}=6x^{3}-125$$

Subtract 6x^{3} from both sides.

$$x^{6}-6x^{3}=-125$$

Add 125 to both sides.

$$x^{6}-6x^{3}+125=0$$

Substitute t for x^{3}.

$$t^{2}-6t+125=0$$

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, -6 for b, and 125 for c in the quadratic formula.

$$t=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 1\times 125}}{2}$$

Do the calculations.

$$t=\frac{6±\sqrt{-464}}{2}$$

Solve the equation t=\frac{6±\sqrt{-464}}{2} when ± is plus and when ± is minus.

$$t=3+2\sqrt{29}i t=-2\sqrt{29}i+3$$

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

$$x=\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}} x=\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}} x=\sqrt{5}e^{\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}} x=\sqrt{5}e^{-\frac{\arctan(\frac{2\sqrt{29}}{3})i}{3}} x=\sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+4\pi i}{3}} x=\sqrt{5}e^{\frac{-\arctan(\frac{2\sqrt{29}}{3})i+2\pi i}{3}}$$