Determine which quadratic congruences have solutions

Question

I need to determine which congruences of the form $ax^2+bx+c\equiv0\pmod{2}$ have solutions. What I know is that $a,b,c$ are all odd. I admit I have no clue how to begin on this one. This is at the end of a chapter on quadratic residues, so I'm supposed to use those to find my results. I know how to deal with stuff of the form $x^2\equiv a\pmod{p}$, but not sure how to apply that to the exercise.

Answer 1

If $a,b,c$ are all odd then $ax^2+bx+c \equiv x^2+x+1 \pmod 2$. Testing the two elements in $\mathbb{F}_2$, we can see there are no solutions.

Generally, for odd primes, we can use the quadratic formula since $2^{-1}$ exists modulo $p$ for odd prime. Then $ax^2+bx+c \equiv 0 \pmod p$ has a solution iff the discriminant $b^2-4ac$ is a quadratic residue.