Trouble understanding this complex numbers problem

Question

The question reads:

$z^2 + |z|i = i$

I've tried approaching this by subbing in $z = (a + bi), |z| = \sqrt{a^2 + b^2},$ and then equating imaginary components, but that didn't seem to help me find an answer. Any ideas? Thanks in advance.

Answer 1

it gives $$x^2-y^2+i(2xy+\sqrt{x^2+y^2}-1)=0$$ so we get $$x^2-y^2=0$$ and $$2xy+\sqrt{x^2+y^2}-1=0$$

Answer 2

Since $z^2=i-\lvert z\rvert i$, $z^2$ is a pure imaginary number and therefore $z=x+xi$ or $z=x-xi$ for some real number $x$. Can you take it from here?

Answer 3

If you rearrange as $$z^2=i-|z|i=i(1-|z|)$$ you can take the modulus of both sides and solve for $|z|$ noting that this is a non-negative real number. Then substitute for $|z|$ in the original equation and $$z=\pm\sqrt i\cdot \sqrt {1-|z|}$$

(the solution of the equation takes two values, you can use whatever convention you like for the principal value)