For how many values of b mod 55 does the congruence x^2 + x + b = 0 (mod 55) have exactly 2 solutions.

Question

Question: For how many values of b mod 55 does the congruence x^2 + x + b = 0 (mod 55) have exactly 2 solutions? I tried to use quadratic function to solve but really don't get it.

Answer 1

Hint: Given any prime modulus, this equation has exactly $0$, $1$, or $2$ solutions, respectively, based on whether its discriminant, which is $1-4b$, is a quadratic nonresidue (giving $0$), equal to $0$ (giving $1$), or a quadratic residue (giving $2$). The number of solutions it has $\bmod pq$ is simply the product of the number of solutions it has $\bmod p$ and the number of solutions it has $\bmod q$. Can you use this to figure out how many solutions it must have $\bmod p$ and $\bmod q$ and then count from there?