# Homework Help: Modulus Question

1. Sep 25, 2012

### BloodyFrozen

1. The problem statement, all variables and given/known data
Find all $z=a+bi$ such that:
$$|z|=|z^{2}+1|$$

2. Relevant equations

3. 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: Sep 25, 2012
2. Sep 25, 2012

### jbunniii

That should be $-b^2$, not $+b^2$.

3. Sep 25, 2012

### BloodyFrozen

Right. Let me just go fix that.

Edit. Fixed

4. Sep 25, 2012

### jbunniii

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

5. Sep 25, 2012

### BloodyFrozen

I fixed then end before, I just forgot to change the middle parts.