X^6-X^3=2

$${\mathtt{y}} = {{\mathtt{x}}}^{{\mathtt{3}}}$$

$${{\mathtt{y}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{y}} = {\mathtt{2}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{y}} = {\mathtt{2}}\\
{\mathtt{y}} = -{\mathtt{1}}\\
\end{array} \right\}$$

$${{\mathtt{x}}}^{{\mathtt{3}}} = {\mathtt{2}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\frac{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\mathtt{0.629\: \!960\: \!524\: \!947\: \!436\: \!6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.091\: \!123\: \!635\: \!972\: \!428\: \!7}}{i}\\
{\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{0.629\: \!960\: \!524\: \!947\: \!436\: \!6}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1.091\: \!123\: \!635\: \!972\: \!428\: \!7}}{i}\right)\\
{\mathtt{x}} = {\mathtt{1.259\: \!921\: \!049\: \!894\: \!873\: \!2}}\\
\end{array} \right\}$$

$${{\mathtt{x}}}^{{\mathtt{3}}} = -{\mathtt{1}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{3}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
{\mathtt{x}} = -{\mathtt{1}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.866\: \!025\: \!403\: \!785}}{i}\right)\\
{\mathtt{x}} = {\frac{{\mathtt{1}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.866\: \!025\: \!403\: \!785}}{i}\\
{\mathtt{x}} = -{\mathtt{1}}\\
\end{array} \right\}$$

So the real solutions are:

• $${\mathtt{x}} = {{\mathtt{2}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3}}}}\right)}$$
• $${\mathtt{x}} = -{\mathtt{1}}$$

Equation  X^6-X^3=2

$$\small{\text{$
\begin{array}{rcl}
x^6-x^3&=&2 \qquad \mathrm{we ~substitute~and ~set~} z = x^3 \\
\mathrm{then~we~have~} \quad z^2-z &=& 2 \\
z^2-z -2 &=& 0\\
z_{1,2} &=& \dfrac{1\pm \sqrt{1-4\cdot(-2)}}{2} \\
z_{1,2} &=& \dfrac{1\pm \sqrt{9}}{2} \\\\
z_{1,2} &=& \dfrac{1\pm 3 }{2} \\\\
z_1 &=& \frac{1 + 3 }{2} = 2 \\\\
z_2 &=& \frac{1 - 3 }{2} = - 1\\\\\\
x_1 & =& \sqrt[3]{z_1}= \sqrt[3]{2}= 1.25992104989\\\\
\mathbf{x_1} & \mathbf{=}& \mathbf{1.25992104989}\\\\\\
x_2 &=& \sqrt[3]{z_2}= \sqrt[3]{-1}= -1\\\\
\mathbf{x_2} &\mathbf{=}& \mathbf{-1}\\\\
\end{array}
$}}$$

Syllogist is right :))

$$\\X^6-X^3=2\\\\
Let \;Y=X^3\\\\
Y^2-Y=2\\\\
$Completing the square$\\\\
Y^2-Y+\frac{1}{4}=2+\frac{1}{4}\\\\
(Y-\frac{1}{2})^2=\frac{9}{4}\\\\
Y-\frac{1}{2}=\pm\frac{3}{2}\\\\
Y-\frac{1}{2}=\frac{3}{2}\qquad OR \qquad Y-\frac{1}{2}=-\frac{3}{2}\\\\
Y=2\qquad \qquad OR \qquad \qquad Y=-1\\\\
X^3=2\qquad \qquad OR \qquad \qquad X^3=-1\\\\
X=\sqrt[3]{2}\qquad \qquad OR \qquad \qquad X=-1\\\\$$

These are just the real solutions :)

Let me think about the complex solutions

I think they are

$$\\x=\sqrt[3]{2}\;\;e^{\frac{2\pi i}{3}},
\qquad x=\sqrt[3]{2}\;\;e^{\frac{-2\pi i}{3}}, \qquad
x=e^{\frac{-\pi i}{3}}, \quad and \quad x= e^{\frac{\pi i}{3}}$$

I hope I didn't s***w that one up :/