Complex solutions of $(z-1)/(z+1)=z+2$

Question

I am having trouble with the following question: If $z=x+yi$, determine the values of $x$ and $y$ such that $$\frac{z-1}{z+1}=z+2 $$

What I have done so far: I solved for $z$ using the quadratic function on my calculator, and got $z= -1\pm \sqrt 2i$

I'm unsure where to go from here.

Answer 1

That's just a quadratic equation which can be solved exactly the same as was done in high school:

$$\frac{z-1}{z+1}=z+2\implies z-1=z^2+3z+2\implies z^2+2z+3=0$$

and the discriminant

$$\Delta:=b^2-4ac=4-12=-8\implies z_{1,2}=\frac{-2\pm\sqrt\Delta}2=\frac{-2\pm2\sqrt2\,i}2=-1\pm\sqrt2\,i$$

Answer 2

It looks like you've got the answer!

If you have $z=-1 \pm \sqrt{2}i$ and need to find $x,y$ if $z=x + iy$, then $x = -1$ and $y = \pm\sqrt{2}$ by inspection.

Answer 3

On one hand, you wrote $z=x+iy$, and on another, $z=-1\pm i\sqrt2$.

Obviously you can conclude $x+iy=-1\pm i\sqrt2$.

Two complex numbers are equal iff they have the same real and imaginary parts. So

$$\begin{cases}x=-1,\\y=\pm\sqrt2.\end{cases}$$