Q:

Finding complex number defined by 3 equations

Let $z$ be a complex number satisfying
$$\DeclareMathOperator{\Re}{Re}\Re[z^4]=1/2$$
$$z\bar{z}+2|z|-3=0$$
$$\arg z \leq \frac{\pi}{4}.$$

A:

Your second equation says, $\vert z \vert ^{2}+2\vert z\vert -3=0$ so $(|z|+3)(\vert z \vert-1)=0$ since $|z|+3>0$  we must have $|z|=1$, hence $z=e^{i\theta}$ for some $\theta \in [0,2 \pi[$. $z^{4}=e^{i4\theta}$. Re$(z^{4})=\cos(4\theta)$, hence $\cos 4 \theta=\frac{1}{2}$ so $4 \theta=\frac{\pi}{3}$ or $4 \theta=\frac{5\pi}{3}$ hence $\theta=\frac{\pi}{12}$ or $\theta=\frac{5 \pi}{12}$, but only $\theta=\frac{\pi}{12}$  is smaller than $\frac{\pi}{4}$. So $z=e^{i\frac{\pi}{12}}$ (Edited thanks to comments).

A:

Like with many things complex, this is best looked at graphically.
The first equation defines some lines.
The second equation defines a circle.
The third equation defines a wedge.
Look at the intersection.

