find all z

**kezman** · July 21st 2006, 02:13 PM

find all z in C that verify simultaneously:

|z|² + |z| - 2 = 0
and
z^8 + z^6 + z^4 + z^2 = 0

Of the first equation I know that 1 is a root.
And I think i is a solution for the second one.
But I cant find a convincing solution.

**Soroban** · July 21st 2006, 04:49 PM

Hello, kezman!

Find all $z \in C$ that satisfy simultaneously: . $\begin{array}{cc}|z|^2 + |z| - 2\:=\:0\\z^8 + z^6 + z^4 + z^2 \:= \:0\end{array}$

The first equation factors: . $(|z| -1)(|z| + 2)\:=\:0$

And we get: . $|z| = 1$
. . (A magnitude cannot be negative.)

The second equation factors: . $z^2(z^6 + z^4 + z^2 + 1) \:= \:0$

Factor: . $z^2\left[(z^4(z^2 + 1) + (z^2 + 1)\right] \:= \:0$

Factor: . $z^2(z^2 + 1)(z^4 + 1)\:=\:0$

And we have: . $z^2 = 0\quad\Rightarrow\quad z = 0$ . . . which does not satisfy the first equation

But these do: $z^2 = -1\quad\Rightarrow\quad z = \pm i$
And so do these: $z^4 = -1\quad\Rightarrow\quad z \,= \,\pm\sqrt{i} \,= \,\pm\frac{1 + i}{\sqrt{2}}$

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