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## Homework Statement

Find all ##z=a+bi## such that:

[tex]|z|=|z^{2}+1|[/tex]

## Homework Equations

## The Attempt at a Solution

I expanded the components.

[tex]|z|=|z^{2}+1|[/tex]

[tex]z^{2}=a^2-b^2+2abi[/tex]

[tex]\sqrt{a^{2}+b^{2}}=\sqrt{(a^{2}-b^{2}+1)^{2}+(2ab)^{2}}[/tex]

[tex]a^2+b^2=(a^{2}-b^{2}+1)^{2}+(2ab)^{2}[/tex]

[tex]0=a^{4}+2a^{2}b^{2}+b^{4}+a^{2}-3b^{2}+1[/tex]

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

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