Avadhoot's Algebra problem

Algebra Level 4

Consider \(f(x)=ax^2+bx+c\), where \(a\), \(b\) and \(c\) are real numbers and \(f(x)\) is such that it takes real values for all real \( x\) and non-real values for all non-real \(x\). Then which of the following is necessarily true?