Quartic

fcukme

Hey guys, i'm looking for a little help here. I ran into the problem of needing to solve for the roots of a quartic equation. (equation with the highest x power being 4). I know the roots come in pairs, so i either have 2 real/2 complex, all real, all complex. I was wondering of there were any conditions that could allow me to look at the case of only 4 real roots. I know that its linked to sum of co-efficients being negative under square root brackets.

Thanks in advance.

ecksor

You can cook one up quite easily by just multiplying out something like (x-4)(x-3)(x-2)(x-1) = 0 or am I misunderstanding the question?

fcukme

You may be on to something, what i'm looking for is a condition that when holds you are sure to get all real roots and no complex roots.
eg if the equation was ax^4+bx^3+cx^2+dx+e=0
then is there a condition in general terms eg a>0 and b*d<0 that when they hold the only solution is 4 real roots. This maybe easy I just can find it in any maths books.

Son Goku

A generic quartic:


has the following determinant:




If this is zero then the quartic has only real roots.

Basically a quartic rarely has all real roots.

fcukme

Thanks a million!!!!!!!!!!!!!!!