Classifcation of irreducible polynomials

[gazzo]

"Conjecture a classifucation rule for all irreducible polynomials of the form ax^2 + bx + c over the reals. Prove it."

I'm stuck cold at the start. classification rule ?

"Let R be an integral domain.
A nonzero f in R[x] is irreducible provided f is not a unit and in every factorization f = gh, either g or h is a unit in R[x].

So, f in R[x] is reducible over R if it can be factorised as f = gh where g,h are in R[x] with deg(g) < deg(f) and deg(h) < deg(f). And irreducible otherwise."

I have no idea where to start, I tried playing with extensions but that seems pointless in the reals.

For some p, b^2 - 4ac < 0 then p is irreducible over R. But that's not getting me anywhere.

Could someone please just give me a tiny hint/word which may shed a ray of light?

Thanks

 

[member 553083]

The classification rule for all irreducible polynomials of the form ax^2 + bx + c over the reals is that any polynomial where b^2 - 4ac < 0 is irreducible over R. This can be proven by showing that any polynomial with b^2 - 4ac greater than or equal to 0 can be factored using the quadratic formula.