<s>[INST] Solving the equation $-2x^3 +10x^2 -17x +8=(2x^2)(5x -x^3)^{1/3}$ I wanna know how to solve this equation: $-2x^3 +10x^2 -17x +8=(2x^2)(5x -x^3)^{1/3}$
I have some trouble to do that and I'd glad with any help I may get.
 [/INST] Cube both sides and collect terms. You should get
\begin{eqnarray}
512 - 3264 x + 8856 x^2 - 13457 x^3 + 12702 x^4 - 7794 x^5 + 
 3136 x^6 - 844 x^7 + 120 x^8 = 0
\end{eqnarray}
which factorizes into
\begin{eqnarray}
(8 - 17 x + 6 x^2) (64 - 272 x + 481 x^2 - 456 x^3 + 258 x^4 - 
   84 x^5 + 20 x^6) = 0.
\end{eqnarray}
Since this is an $8^{\text{th}}$-degree polynomial equation, you can use Mathematica or a calculator to determine six of the eight solutions (by solving the $6^{\text{th}}$-degree polynomial equation numerically) and the other two by solving the quadratic equation: $\frac{1}{12}(17 \pm \sqrt{97})$. The 6 solutions are 3 pairs of complex conjugates:
\begin{eqnarray}
& & 0.647522 \dots \pm i 2.21209 \dots  \
& & 0.657656 \dots \pm i 0.218497 \dots  \
& & 0.794822 \dots \pm i 0.788971 \dots 
\end{eqnarray}
Of course, you'll have to check that these solutions work. At most one real solution and two complex solutions are spurious.
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