If $\displaystyle z=x+yi$ find the values of $\displaystyle x$ and $\displaystyle y$ such that $\displaystyle \frac{z-1}{z+1}=z+2$
Thanks for your help!
$\displaystyle \frac{x-1+yi}{x+1+yi}=x+2+yi\Rightarrow x-1+yi=x^2+3x-y^2+2+i(2xy+3y)\Rightarrow$
$\displaystyle \Rightarrow\left\{\begin{array}{ll}x^2+3x-y^2+2=x-1\\2xy+3y=y\end{array}\right.\Rightarrow\left\{\be gin{array}{ll}x^2-y^2+2x=-3\\2y(x+1)=0\end{array}\right.$
Now, you have to solve the system. Can you do that?