Noting Roots of a Certain type of Cubic Equation, what if we have the following simpler form for real $d$:
(With $a,b,c\in \mathbb R^+$.) Clearly, depending on $d$, the number of real roots can be changed from $3$ to $1$ easily. Is it possible to get a simple form for the solution(s)?
I got as far as to see that real root $x=k$ is a solution to the following when $0\lt a\le b\le c$ and $d\in\mathbb R^+$:
Every time I attempt to run through the various mechanisms to solve the cubic with these conditions, the number of terms in each expression explodes in a way that makes the effort untenable. It almost seems like this equation should be analyzed in an entirely different way and that it would end up with a different but similar set of solutions relative to the standard set.
Any hints or solutions would be appreciated. Note that I am primarily interested in the real root(s) for $d\in\mathbb R^+$.