I have the following cubic equation
where $A$ is an arbitrary (real) number.
I know that either:
- The 3 roots will be real.
- One root will be real and the other two will be complex conjugates of each other.
I would like to find out
- For what value/values of A the roots change from 3 real roots to one real and two complex roots.
- The signs of each of the real roots (both when they are all real and when there is only one real root)
Is there an analytical way of finding this as a function of $A$ or the only option is to solve the cubic numerically?