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.