Even in the complex case the full solution can be pretty complicated, if you ask for the full solvability conditions;

eqn = (-4 + c1^2 + 2 c1*c2 + c2^2 + (s1 + s2)^2)/4 - 
   3/8*(a2*r1 + a1*r2)*(c2*s1 - c1*s2)*x + 
   9/64*(a1^2*r2^2*(c1^2 + s1^2) - 2*a1*a2*r1*r2)*x^2;
Echo[Reduce[eqn == 0, x]];

In the real case, the solvability conditions involve analyzing a semialgebraic variety of high degree in several variables.

It is true that when the generated conditions are too complicated they are hard to use in practice. Would you like Mathematica to return simply

Echo[Discriminant[eqn, x] >= 0];

as the condition for real solutions?

I know, Gianluca. I prefer Echo since it makes the code and the output more readable. And since I can easily disable it globally by:

$ContextPath = Prepend[$ContextPath, "myContext`"];
Echo[any___] := Null;

Actually I use my own echo-function with takes a global variable testLvl that controls which Echo-output shall be produced.

This works indeed, Mariusz (within 0,06 secs). But why? I had never thought that solving in Complex would be easier than within Reals. But anyway this is a valid solution. I can apply the Reals-condition later to the result.