<s>[INST] How to find all solutions for $x^3=6x+6$ Could anyone help me to find how to find all solution for $x^3=6x+6$?
 [/INST] Set $x=ay$; the equation becomes $a^3y^3-6ay=6$. We want that
$$
\frac{a^3}{6a}=\frac{4}{3}
$$
so we can take $a=2\sqrt{2}$. Then we have
$$
16\sqrt{2}y^3-12\sqrt{2}y=6
$$
that becomes
$$
4y^3-3y=\frac{3}{4}\sqrt{2}>1
$$
Now set $y=\cosh z$, so the equation becomes
$$
4\cosh^3z-3\cosh z=\frac{3}{4}\sqrt{2}
$$
that is,
$$
\cosh3z=\frac{3}{4}\sqrt{2}
$$
Now solve
$$
e^{3z}+e^{-3z}=\frac{3}{2}\sqrt{2}
$$
and you'll have $z$ and so $y$ and finally $x$.
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