Find all the values of $w$ ∈ C that satisfy the equation.

Find all values of $\omega \in \mathbb{C}$ such that $\frac{\omega – \frac1\omega}{2i} = 2.$

So what I have is:

Let $\omega$ be represented as $x + yi$ (by definition of complex numbers)

Then $\frac1\omega = \frac{x-yi}{x^2 + y^2}$ (by properties of complex numbers)

So, $\frac{x+yi – \frac{x-yi}{x^2 + y^2}}{2i} = 2$

Eventually, I get, via expansion and simplification:

$\frac{x(x^2 + y^2 – 1) + yi(x^2 + y^2 + 1)}{2i(x^2 + y^2)}$

Any hints on how I can continue? I feel like I’m close but missing something.

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Note that

$$\frac{\omega – \frac1\omega}{2i} = 2\iff \omega – \frac1\omega=4i \stackrel{\omega \ne 0}{\iff} \omega^2-1=4i\omega\iff\omega^2-4i\omega-1=0.$$

This equation should be much more familiar.

This is an ordinary algebraic equation in the complex variable $\omega$, and in cases like this, it’s almost always not wise to look at the real and imaginary parts of $\omega$.

Just pretend you’re in high-school and solve:
\begin{align}
\omega + 1/\omega&=4i\\
\omega^2-4i\omega-1&=0\\
\omega=2i\pm\sqrt{-3}\\
\omega=(2\pm\sqrt3)i
\end{align}