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] <A HREF="http://www.sciencedirect.com/science/article/pii/S0047259X9691653X">The Probability that a Random Real Gaussian Matrix has k Real Eigenvalues, Related Distributions, and the Circular Law</A>, A. Edelman (1996).