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 ${\rm det}\,M=c$.