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well, to find a "natural way" to distribute the coefficients $b,c$ in the plane, 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 circular law and the probability that a random matrix has k real eigenvalues, A. Edelman (1993). (Published in Journal of Multivariate Analysis 60 (1997) 203-232.)

well, to find a "natural way" to distribute the coefficients $b,c$ in the plane, 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 real roots is $1/\sqrt{2}$.

[1] The circular law and the probability that a random matrix has k real eigenvalues, A. Edelman (1993). (Published in Journal of Multivariate Analysis 60 (1997) 203-232.)

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 circular law and the probability that a random matrix has k real eigenvalues, A. Edelman (1993). (Published in Journal of Multivariate Analysis 60 (1997) 203-232.)

well, to find a "natural way" to distribute the coefficients $b,c$ in the plane, 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 circular law and the probability that a random matrix has k real eigenvalues, A. Edelman (1993). (Published in Journal of Multivariate Analysis 60 (1997) 203-232.)

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 circular law and the probability that a random matrix has k real eigenvalues, A. Edelman (1993). (Published in Journal of Multivariate Analysis 60 (1997) 203-232.)

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 circular law and the probability that a random matrix has k real eigenvalues, A. Edelman (1993). (Published in Journal of Multivariate Analysis 60 (1997) 203-232.)