quadratic one

roots are a and a²

a + a² = -p/3 ...(1)

a³ = 1

so roots are ω and ω² .. cube roots of unity

we know 1 + ω + ω² = 0

now, cube eqⁿ (1)

a³ + a⁶ + 3a⁴ + 3a⁵ = - p³/27

2 + 3( a + a² ) = - p³/27

2 + (3 x -1) = - p³/27

p³ = 27

So, p = 3

@mirka...
once you get that a=ω and ω², the relation
a+a²=-p/3 implies that p=3 since we must have 1+a+a²=0.
There is no need of cubing the equation and doing whatever u did.