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 $\sqrt{x^2 + y^2}.$