Suppose we have a cubic equation $$ ax^3 +bx^2 +cx +d =0 $$ for which we know that all three distinct roots are real. Do we have a formula for them that does not involve complex roots of unity?
The reason I'm asking is because my equation has three distinct roots, but it is highly parametrized and it is hard to see that all three roots obtained by Caradano's formula are real.
For such a case, there is the trigonometric method