How can we solve the expression explicitly for $X$ in terms of $Y$? I am thinking how to write $X$ explicitly in terms of $Y,A,B,C$?
I have $AX^3 + X^2(B-1) + X(-C) + \alpha = Y$
I thought of using symbolic maths but could not find any.
Any help is nice!
 A: For the solutions of cubic equation, you eed to follow the steps described here.
Applied to the case of your equation
$$A x^3 +(B-1)x^2 -C x + (\alpha-y) = 0$$ because the formula are quite nasty, I strongly suggest that you start defining some intermediate coefficients such as, for example,
$$b=\frac{B-1}A \qquad c=-\frac C A\qquad d=\frac {\alpha-y} A$$ (assuming $A \neq 0$) to make the equation
$$x^3+b x^2+c x+d=0$$
Now, you will need to compute the key quantities
$$\Delta =b^2 c^2+18 b c d-4 c^3-27 d^2-4 b^3 d$$
$$p=c-\frac{b^2}{3}$$
$$q=\frac{2 b^3}{27}-\frac{b c}{3}+d$$ and follow the steps given in the linked page.
If you are only concerned by the possible single real solution (this implies $\Delta <0$), use what is given in the "hyperbolic solution for one real root" paragraph. It could be a nasty formula but totally workable.
If you face three real solutions  (this implies $\Delta >0$), use what is given in the
"trigonometric solution for three real roots" paragraph. It could be a nasty formula but totally workable.
