I am trying to find the roots of a cubic polynomial in variable $r$: $ar^3 - r^2 +2mr -P^2=0$, $P$, $m$ and $a$ are constants here.
I know that the discriminant of this polynomial for cubic roots is: $4 M^2 - 4 P^2 - 32 M^3 a + 36 M P^2 a - 27 P^4 a^2$
Could some one help me in finding the roots of this polynomial by using this discriminant. I want some sophisticated formula for this purpose, just like we have quadratic formula for quadratic polynomial (unluckily I cant find it on internet such that the roots do not involve trigonometric functions).
Further I want to know the nature of the roots of a cubic polynomial if the discriminant is less than zero.
Thanking in advance.