assumptions – Solving an Algebraic Equation with Constraints on Multiple Parameters


Suppose we have the following equation to solve (with respect to $x$):

$$
frac{momega^2x^2}{2} + frac{malpha x^3}{3} = E,quadtext{where }m,omega,alpha,E>0text{ and } E < frac{momega^6}{6alpha^2}.
$$

Now, theoretically speaking, along with these constraints, we should have derived 3 real roots for this equation. But when I tried to actually solve it on Mathematica, whether I had added these constraints in the Assumptions field, it kept giving me the complex roots. What am I supposed to do to let Mathematica derive the desired real roots? Any of your help will be greatly appreciated! 🙂