is this method for solving a complex equation correct?

Question

I need to find roots of this complex equation: $$\left(|z|^2 + 3 |z|\right)^2 \left(z^8 - 1\right)^3 = 0$$ (for the moment, let's focus only on the first parenthesis) $$\left(|z|^2 + 3 |z|\right)^2 = 0$$

My attempt: I collected $|z|$, because it's a common term.
I got $\sqrt{a^2 + b^2} = 0$, and the other expression is always false, because norm is always positive. the first one could be rewritten as $a^2 + b^2 = 0$, and this is the equation of a circle of radius = $0$. so it's a point (i.e the origin).

does this method make sense? is it correct? Or, should I use : $$\rho = |z| = \sqrt{a^2 + b^2}$$ instead?

Answer 1

Well, as you wrote:

$$\underbrace{\Re\left(\text{z}\right)^2+\Im\left(\text{z}\right)^2}_{:=\space x}+3\sqrt{\underbrace{\Re\left(\text{z}\right)^2+\Im\left(\text{z}\right)^2}_{=\space x}}=0\tag1$$

So, we get:

$$x+3\sqrt{x}=0\space\Longleftrightarrow\space\sqrt{x}\left(3+\sqrt{x}\right)=0 \space\Longrightarrow\space x=0\tag2$$

And solving:

$$\text{z}^8=1\space\Longleftrightarrow\space\text{z}=\exp\left(\frac{\pi \text{k}}{4}\cdot i\right)\tag3$$

With $\text{k}\in\mathbb{Z}$ and $0\le\text{k}\le7$.