well, you could treat this problem as the special case $n=2$ of a classic problem in random-matrix theory: take an $n\times n$ real matrix $M$ with independent identical normal distributions of the matrix elements $M_{nm}$; what is the probability that all $n$ eigenvalues are real?
for $n=2$ we would then identify ${\rm tr}\,M=-2b$ and ${\rm det}\,M=c$.
this probability is known exactly [1] for any $n$, it equals $2^{-n(n-1)/4}$; so for $n=2$ the probability of complex roots is $1-1/\sqrt{2}$.
[1] The Probability that a Random Real Gaussian Matrix has k Real Eigenvalues, Related Distributions, and the Circular Law, A. Edelman (1996).