<s>[INST] Problem getting the real roots of this complex expression I'm trying to get the real roots of this expression:
$$\dfrac{1}{z-i}+\dfrac{2+i}{1+i} = \sqrt{2}$$
Where $i^2=-1$ and $z=x+iy$.
I tried to simplify that with Algebra, and then separate the real and imaginary parts in both sides of the expression to obtain an equation system, so I would solve it to obtain the roots for both $x$ and $y$. But all I get is a mess!
Any help would be appreciated, thank you! :)
P.S. It comes again from a Russian book, it says the answer is: there aren't real solutions. And with the procedure I said, I got real solutions!
P.P.S. I'd write down what I did, but I don't have the written steps anymore, sorry :(
 [/INST] This can be done by "getting the unknown alone" by simplifying the fraction and undoing each operation on the left side.
We get
$$\begin{gathered}
  \frac{1}{{z - i}} + \frac{{2 + i}}{{1 + i}} = \sqrt 2  \\
  \frac{1}{{z - i}} + \frac{3}{2} - \frac{1}{2}i = \sqrt 2  \\
  \frac{1}{{z - i}} = \sqrt 2  - \frac{3}{2} + \frac{1}{2}i \\
  z - i =  - \frac{1}{3} + \left( { - \frac{2}{3}\sqrt 2  - 1} \right)i \\
  z =  - \frac{1}{3} - \frac{2}{3}\sqrt 2 i \\ 
\end{gathered} $$
As you see, there is no real solution: just one complex one. I left out the hairy division and reciprocal steps: let me know if you need them.

Here are more details on the first division:
$$\begin{gathered}
  \frac{{2 + i}}{{1 + i}} \\
   = \frac{{(2 + i)(1 - i)}}{{(1 + i)(1 - i)}} \\
   = \frac{{2 - 2i + i - {i^2}}}{{1 - {i^2}}} \\
   = \frac{{2 - 2i + i + 1}}{{1 + 1}} \\
   = \frac{{3 - i}}{2} \\
   = \frac{3}{2} - \frac{1}{2}i \\ 
\end{gathered} $$
And here is the reciprocal:
$$\begin{gathered}
  \frac{1}{{\sqrt 2  - \frac{3}{2} + \frac{1}{2}i}} \\
   = \frac{{\sqrt 2  - \frac{3}{2} - \frac{1}{2}i}}{{\left( {\sqrt 2  - \frac{3}{2} + \frac{1}{2}i} \right)\left( {\sqrt 2  - \frac{3}{2} - \frac{1}{2}i} \right)}} \\
   = \frac{{\sqrt 2  - \frac{3}{2} - \frac{1}{2}i}}{{{{\left( {\sqrt 2  - \frac{3}{2}} \right)}^2} - {{\left( {\frac{1}{2}i} \right)}^2}}} \\
   = \frac{{\sqrt 2  - \frac{3}{2} - \frac{1}{2}i}}{{\left( {2 - 3\sqrt 2  + \frac{9}{4}} \right) + \frac{1}{4}}} \\
   = \frac{{\sqrt 2  - \frac{3}{2} - \frac{1}{2}i}}{{\frac{9}{2} - 3\sqrt 2 }} \\
   = \frac{{\left( {\sqrt 2  - \frac{3}{2} - \frac{1}{2}i} \right)\left( {\frac{9}{2} + 3\sqrt 2 } \right)}}{{\left( {\frac{9}{2} - 3\sqrt 2 } \right)\left( {\frac{9}{2} + 3\sqrt 2 } \right)}} \\
   = \frac{{\frac{9}{2}\sqrt 2  + 3 \cdot 2 - \frac{{27}}{4} - \frac{9}{2}\sqrt 2  - \frac{9}{4}i - \frac{3}{2}\sqrt 2 i}}{{\frac{{81}}{4} - 9 \cdot 2}} \\
   = \frac{{ - \frac{3}{4} + \left( { - \frac{3}{2}\sqrt 2  - \frac{9}{4}} \right)i}}{{\frac{9}{4}}} \\
   =  - \frac{1}{3} + \left( { - \frac{2}{3}\sqrt 2  - 1} \right)i \\ 
\end{gathered} $$
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