Suppose that a+bi is a square root of 5 - 12i. 

Then, (a+bi)^2 = (a^2 - b^2) + (2ab)i = 5 - 12i. 



Equate real and imaginary parts: 

a^2 - b^2 = 5 

2ab = -12 ==> b = -6/a. 


So, a^2 - (-6/a)^2 = 5 

==> a^2 - 36/a^2 = 5 

==> a^4 -5a^2 - 36 = 0. 

==> (a^2 -9)(a^2 + 4) = 0. 

Since a must be real, a = 3 or -3. 

This gives b = 2 or -2, respectively. 



Thus, we have two square roots: 3-2i or -3+2i.