Criterion for a splitting field of a quadratic polynomial to be a Galois Extension

Question

Let $F$ be a field of characteristic, $f(x) = x^2 + bx + c \in F[x]$ an irreducible polynomial. I have to give a criterion for the splitting field $E$ of $f$ over $F$ to be a Galois extension.

Now, we essentially want the polynomial to be separable. Therefore, we want $D(f) \neq 0 \implies b^2 \neq 4c$

And for the case $\text{char }F=2$, we want $b \neq 0$, and if $b=0$, then we want $c$ to be a non-square in $E$

Is this criterion exhaustive?