If $\left|z+\frac{2}{z}\right |=2$, then prove that max value of $|z|$ is $\sqrt 3 + 1$

Question

Solving the quadratic inequality

$$|z|\ge1+i$$ and $$|z|\le1-i$$

I don’t know how to solve further. Please help

Thanks!

Answer 1

It should read $$|z|-\frac 2{|z|}=2$$ since the minimal polynomial of $\sqrt 3 +1$ is $X^2-2X-2$.

Now with the correct question at hand, simply note that since every complex number appears inside a modulus, this question really is about non-negative real numbers. Solving the quadratic equation tells you what the possibilities are, and $\sqrt 3 +1$ is easily seen to be the largest (and also the only one that is non-negative).

Therefore $\sqrt 3+1$ is not simply the largest possible value of $|z|$, it is the only possible value.

Answer 2

It should read $|z|-\frac{2}{|z|}=2$. If $t:=|z|$ , we get the equation $t^2-2t-2=0.$ This equation has the sulutions $t_{1/2}=1 \pm \sqrt{3}.$ Can you proceed ?

Edit: your questions has changed !