  Quadratic functions have applications in cryptography. In this paper, we
investigate the modular quadratic equation $$ ax^2+bx+c=0 \quad (mod \,\, 2^n),
$$ and provide a complete analysis of it. More precisely, we determine when
this equation has a solution and in the case that it has a solution, we not
only determine the number of solutions, but also give the set of solutions in
$O(n)$ time. One of the interesting results of our research is that, when this
equation has a solution, then the number of solutions is a power of two.
