for a given range of x in Zn , and n is composite , and ax² + bx + c ≡ 0(mod n) and if (4a,n)=1,

I learned that we can solve the congruence by (2ax + b)² ≡ b²-4ac (mod n) ==> y² ≡ z (mod n)

So, if n is composite,

Sometimes I see, modulo 4an, when do we take 4an and n ,

how can we prove , there exists residues and non-residues as z values. for any range of x taken from Zn

Is there any range of x in general , such that there exists only either residues or non residues as solutions.

If i am wrong or obscure any where in my question , hope will be notified to me.