<s>[INST] How to solve this equation if $x\in \mathbb{R}$, $x ^3+1 =2\sqrt [ 3 ]{ 2x-1 }$, find $x$ 
If $x\in \mathbb{R}$, and $$x ^3+1 =2\sqrt [ 3 ]{ 2x-1 }$$ find $x$.

I asked my teacher to help me in solving it, he responded I can't! So, I hope you help me in solving it or even giving me hints that will help me in solving.
 [/INST] By factoring we see that
$$
x^9+3x^6+3x^3-16x+9=(x^6 + 2x^4 + 2x^3 + 4x^2 + 2x + 9)(x^2 + x - 1)(x - 1),
$$
so that we have solutions $x=1$, $x=\frac{\pm \sqrt{5}-1}{2}$ and $6$ further complex solutions. So we have exactly three real solutions, easy to compute.
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