# |z⁴| + 1 - 2(z*)²= 0

## 2 Answers

Note that $|z|$ is a real number, so if the equation $|z⁴| + 1 - 2(z^*)²= 0$ is satisfied, $(z^*)^2$ should also be a real number. This means that $z^*$ must be either real or purely imaginary, i.e. $z=x$ or $z=ix$ for some real number $x$.

I) If $z=x$. Then the equation becomes

\[x^4-2x^2+1=0 \Rightarrow (x^2-1)^2=0 \Rightarrow x=\pm 1.\]

I) If $z=xi$. Then the equation becomes

\[x^4+2x^2+1=0 \Rightarrow (x^2+1)^2=0 \Rightarrow \text{no solution}.\]

So $x=x=\pm 1$ are the only solutions.

You can use this expression $a^2 - b^2 = (a+b)(a-b)$ by taking either $a = +z^2$ or $a=-z^2$ and $b=z$.

For $a = +z^2$ you will end up with $(z^2+z)(z^2-z) + 1 - z^2 = 0$ and solving this will give $z=-1$

Repeat for $a=-z^2$ and it will give $z=1$

Hope it helps :)

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