Characterization of integers which has a $2$-adic square root
Solution 1:
You are looking for $x\in\mathbb Z_2$ such that $(1+2x)^2=8b+1$. This is equivalent to $$ x^2+x-2b=0.$$ Now mod $2$, this polynomial is $x(x+1)$ and has (at least) one simple root in $\mathbb F_2$. By Hensel's lemma, the above equation has a solution in $\mathbb Z_2$.