Let z ∈ C such that z^2+3i/z-2+i = 2+3i ​ then sum all possible values of z^2 is ​

Question

Let \(z \in C\) such that \(\frac{z^2+3i}{z-2+i}\) = 2 + 3i then sum all possible values of \(z^2\) is

(1) -19 - 2i

(2) 19 + 2i

(3) -19 + 2i

(4) 19 - 2i

Answer 1

Correct option is: (1) -19 - 2i 

Which has roots as \(\alpha\) and \(\beta\) sum of whose squares will be given as

\(\alpha^2 + \beta^2=(\alpha+\beta)^2-2\alpha\beta\) 

\(= (2+3i)^2 = -2 \times 7(1+i)\) 

\(= (4-9+12i)-14-14i\)  

\(= -19 -2i\)