Quadratic inequality with a parameter

Question

I am trying to solve a quadratic inequality with a parameter. I feel dumb as I don't know how to proceed. The inequality is as follows:

$(a + c_1)^2 \le c_2$, where $c_1$, $c_2$ are constants and $c_2$ is negative and $a$ is a parameter of choice. Is there an analytical solution?

Answer 1

I can give you the Mathematica output. If you allow a to be complex and c2<0:

Reduce[(a + c1)^2 <= c2 && c2 < 0, a, Complexes]

c2 < 0 && Re[a] == -Re[c1] && 
(Im[a] <= -Im[c1] - Sqrt[-c2 + Re[a]^2 + 2 Re[a] Re[c1] + Re[c1]^2] || 
 Im[a] >= -Im[c1] + Sqrt[-c2 + Re[a]^2 + 2 Re[a] Re[c1] + Re[c1]^2])

You get that the real part of a and c1 have to cancel and some conditions on the imaginary parts of a and c1.