<s>[INST] How can I find maximum and minimum modul of this complex? Let be given the complex $z$ such that $$\left |z + \dfrac{1}{z}\right |=3.$$ Find  maximum and minimum modul of complex $z$.
I tried. Put $z=x+i y$. From $$\left |z + \dfrac{1}{z}\right |=3.$$ We have $z^2 + 3z|$ = 3|z|.$ Threrefore 
$$1 + 2 x^2 + x^4 - 2 y^2 + 2 x^2 y^2 + y^4 - 9 (x^2 + y^2)=0.$$
From here, I cann't find max and min of the $\sqrt{x^2 + y^2}.$
 [/INST] Hint. By using samjoe's comment $z\bar{z}=|z|^2$, we have that the given equation is equivalent to
$$|z|^2+2\cos(\theta)+\frac{1}{|z|^2}=\left |z + \dfrac{1}{z}\right |^2=3^2=9, \;\;|z|^2+\frac{1}{|z|^2}=9-2\cos(\theta)$$
where $z=|z|(\cos(\theta)+i\sin(\theta))$. Note that
$$7=9-2\cos(0)\leq|z|^2+\frac{1}{|z|^2}\leq 9-2\cos(\pi)= 9+2=11.$$
Now try to show that the maximum of $|z|$ is attained when
$$|z|^2+\frac{1}{|z|^2}=11,$$
that is for $|z|=(11+3\sqrt{13})/2$.
What about the minimum?
</s>