How do I write explicitly the roots of $x^3 - x+ \eta =0$? I tried online calculators but could not get any idea, also doing $x(x-1)(x+1) = -\eta$ seems is not helping too.
I tried to cleverly see for any one root since I after getting one root I can do the quadratic solving for other two roots but getting that one root is difficult as I see
Let
$$x=\frac2{\sqrt3}\cos\theta.$$
Then
$$\frac8{3\sqrt3}\cos^3\theta-\frac2{\sqrt3}\cos\theta+\eta=0,$$
$$4\cos^3\theta-3\cos\theta+\frac{3\sqrt3}2\eta=0,$$
so
$$\cos3\theta=-\frac{3\sqrt3}2\eta$$ gives you 3 solutions.
When
$$\left|\frac{3\sqrt3}2\eta\right|>1,$$ just use
$$x=-\text{sgn }\eta\,\frac2{\sqrt3}\cosh\theta,\\\cosh3\theta=\frac{3\sqrt3}2|\eta|.$$