Solving complex variable equation

Question

I am in my first Linear Algebra course and, until now, I felt good. I've been working on the following exercise for one entire day, to no avail.

Find al Z such that:

$$|z|^{2} = 3z^{2}+i\cdot z^{*}$$

It is the first complex equation we were given to solve, and I honestly dont know where should I start from.

Could anyone give me a hint or a tip, just a place to "pull the string from"?

Lots of thanks in advance.

Answer 1

Let $z= x + i y$. Then \begin{align} |z|^2 & = 3z^2 + i z^* \\ \implies x^2+y^2 &= (3x^2-3y^2 +y) + i(6xy+x) \\ \implies x(6y+1) & = 0 \\ \vdots \end{align}

Answer 2

$$ z \cdot z^* = 3z^2+z^*i$$ so $$ z^* = {3z^2\over z-i}\;\;\;\;(1)$$ and

$$ z = {3{z^*} ^2\over z^*+i}\;\;\;\;(2)$$

Now plug (1) in (2)....

Answer 3

Hint...Try substituting $z=a+ib$ into the equation and compare real and imaginary parts