Solving Modulus Equation: Find z=a+bi

[BloodyFrozen]

Homework Statement

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

Homework Equations

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...

 

[jbunniii]

Homework Statement

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

Homework Equations

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 $+b^2$.

 

[BloodyFrozen]

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

Right. Let me just go fix that.

Edit. Fixed

 

[jbunniii]

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.

 

[BloodyFrozen]

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.