# Formula for finding the Roots of a cubic polynomial and nature of roots depending on the discreminant

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.

• @iadvd Is there a reason you are digging up old posts and making minor edits? This is unfortunate because, as mentioned before, all of your edits must be approved until you hit 2000 rep. Your edits also mean that old, inactive posts are bumped to the front page, pushing out new, relevant questions. Please consider waiting until you reach 2000 rep to make such editing decisions when you edits will be applied immediately. Edits are meant to be substantive, not simple "gimme +2 rep" tools. – Daniel W. Farlow Apr 3 '15 at 5:19
• @crash sorry I will stop reviewing... I am just reading old posts, because there is not much recent updates where to help, just to get used to Mathjax and push some unanswered questions. Last time you will see it, I promise. – iadvd Apr 3 '15 at 5:21
• @iadvd You misunderstand me. Your reviews are valued. They are. I myself have made numerous reviews and have edited old posts. I'd say keep to editing newer posts right now, but please don't think at all that you should not improve posts when you see fit. :) – Daniel W. Farlow Apr 3 '15 at 5:23
• @crash thank you! I will do so. :) – iadvd Apr 3 '15 at 5:24