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Math Help - Please help me!! >.<

  1. #1
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    Question Please help me!! >.<

    Please teach me how to solve this question...

    Two complex number w and z are such that w*= z -2i and |w|^2=z+6. By eliminating z, find w in the form a+ib, where a and b are real and positive.

    please help me! I really have many problems with complex number!!
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  2. #2
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    Re: Please help me!! >.<

    I assume you are using the * to represent conjugate. You should know that \displaystyle \begin{align*} w\,\overline{w} = |w|^2 \end{align*}, so we have \displaystyle \begin{align*} w\,\overline{w} = z + 6 \end{align*} and \displaystyle \begin{align*} \overline{w} = z - 2i \end{align*}. Subtracting the second equation from the first gives

    \displaystyle \begin{align*} w\,\overline{w} - \overline{w} &= 6 + 2i \\ \overline{w}\left( w - 1 \right) &= 6 + 2i \\ \left( a - b\,i \right) \left( a - 1 + b\,i \right) &= 6 + 2i \\ a(a - 1) + ab\,i - (a - 1)b\,i - b^2 i^2 &= 6 + 2i \\  a^2 - a + b^2 + b\,i &= 6 + 2i \\ a^2 - a + b^2 = 6 \textrm{ and } b &= 2 \\ a^2 - a + 4 &= 6 \\ a^2 - a - 2 &= 0 \\ (a - 2)(a + 1) &= 0 \\ a = 2 \textrm{ or } a &= -1 \end{align*}

    So the solution is \displaystyle \begin{align*} w = 2 + 2i \end{align*} or \displaystyle \begin{align*} w = -1 + 2i \end{align*}
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  3. #3
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    Re: Please help me!! >.<

    Are you sure you copied the problem correctly?
    Assuming you did:
    1. What does |w|^2=z+6 say about the complex number z? (Hint: what kind of values can |w| take?)
    2. Do you know "eliminate z" algebraically? (If not, then that kind of algebra is something you should heavily review - it's very important.)
    3. Just so you can see how you're doing - it's eventually a quadratic you'll need to solve, and so you'll have two solutions for w. And they'll be clean, like (integer) + i(integer).
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