Let $f_1,\cdots,f_r$ be a set of monic irreducible polynomials of degree $2$ in $\Bbb Z[x]$. Can we find a prime number $p$ such that no $f_i$ has a root in $\Bbb F_p$?
I have thought about the density, and I know that: for any $i$, the set $p$ such that $f_i$ splits in $\Bbb F_p$ has density $\frac{1}{2}$. But the density is too large, so I don't know how to get the result.
I'm OK with algebraic number theory and basic analytic number theory, hence the methods using these two are welcome.