Say $f(x) = ax^2 + bx + c$ with $\gcd(a,b,c) = 1$. So I have read a little literature and the approach to determining whether $f(x)\equiv 0\pmod {2^k}$ exists $\forall k\in\mathbb N$ seems to focus on parity of $a$, $b$ and $c$ modulo $2$.
I also read that out of $7$ possible parities, $5$ of these parities have been successfully worked on $3$ out of $5$ gives that $f(x)$ has a root modulo all powers of $2$; other two gives that $f(x)$ does not have roots modulo all powers of $2$.
So only possible cases seem to be left are when either $a$ and $c$ are odd and $b$ even or $a$ is odd and $b$, $c$ are even. I have two questions:
Is this problem open or solved? I could not find any result based on two cases.
What are the repercussions of successfully dealing with this problem, specifically the last two cases? Would this imply anything that is already not obvious in the problem statement?