# Modulus Question

## Homework Statement

Find all $z=a+bi$ such that:
$$|z|=|z^{2}+1|$$

## The Attempt at a Solution

I expanded the components.
$$|z|=|z^{2}+1|$$
$$z^{2}=a^2-b^2+2abi$$
$$\sqrt{a^{2}+b^{2}}=\sqrt{(a^{2}-b^{2}+1)^{2}+(2ab)^{2}}$$
$$a^2+b^2=(a^{2}-b^{2}+1)^{2}+(2ab)^{2}$$
$$0=a^{4}+2a^{2}b^{2}+b^{4}+a^{2}-3b^{2}+1$$

I don't see what to do now...

Last edited:

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jbunniii
Homework Helper
Gold Member

## Homework Statement

Find all $z=a+bi$ such that:
$$|z|=|z^{2}+1|$$

## The Attempt at a Solution

I expanded the components.
$$|z|=|z^{2}+1|$$
$$z^{2}=a^2+b^2+2abi$$
That should be $-b^2$, not $+b^2$.

That should be $-b^2$, not $+b^2$.
Right. Let me just go fix that.

Edit. Fixed

jbunniii
Homework Helper
Gold Member
Right. Let me just go fix that.

Edit. Fixed
Well, you fixed it in the first line where it appeared, but you still need to fix it everywhere else.

Well, you fixed it in the first line where it appeared, but you still need to fix it everywhere else.
I fixed then end before, I just forgot to change the middle parts.