# Thread: Prove general solution for deppresed cubic equation y^3+py+q=0 given a discriminant

1. ## Prove general solution for deppresed cubic equation y^3+py+q=0 given a discriminant

I've been given the following problem and I can't seem to figure out the solution, would appreciate any help and direction to solution !

Let's look at the equation : y3 + py + q = 0 (*)
We shall define Delta (or d in short) d = 4p3 + 27q2
Prove that :
a) If d > 0 , then (*) has a single solution .

b) If d = 0 , and at least one of the coefficients (p , q) =/= 0, then (*) has 2 solutions .

c) If d < 0 , then (*) has 3 solutions .

Thank you very much in advance !

2. ## Re: Prove general solution for deppresed cubic equation y^3+py+q=0 given a discrimina

if $r$ is a real root then

$y^3+p y + q=(y-r)\left(y^2+r y+r^2+p\right)$

the existence of additional real roots depends on the discriminant of the quadratic factor

use the identity

$\left(3r^2+4p\right)\left(p+3r^2\right)^2=4 p^3+27 q^2$