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Math Help - find all z

  1. #1
    Member kezman's Avatar
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    find all z

    find all z in C that verify simultaneously:


    |z| + |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.
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  2. #2
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    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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