Solving Modulus Equation: Find z=a+bi

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SUMMARY

The discussion focuses on solving the modulus equation |z|=|z²+1| for complex numbers z=a+bi. Participants detail their attempts to expand the equation, leading to the expression a²+b²=(a²-b²+1)²+(2ab)². A critical correction was made regarding the sign of the b² term in the expansion, which is essential for accurate calculations. The final equation simplifies to 0=a⁴+2a²b²+b⁴+a²-3b²+1, indicating the need for further analysis to find solutions.

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  • Understanding of complex numbers and their representation as z=a+bi
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  • Experience with algebraic manipulation and simplification techniques
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  • Learn techniques for solving polynomial equations, particularly quartic equations
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Students studying complex analysis, mathematicians tackling polynomial equations, and educators seeking to enhance their understanding of modulus equations in complex numbers.

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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...
 
Last edited:
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BloodyFrozen said:

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.
 
jbunniii said:
That should be -b^2, not +b^2.

Right. Let me just go fix that.

Edit. Fixed
 
BloodyFrozen said:
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.
 
jbunniii said:
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.
 

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