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Math Help - find complex numbers

  1. #1
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    find complex numbers

    find all complex numbers, z, which satisfy the equation z^2 = 6 -8i

    do i begin by writing z^2 - 6 - 8i = 0 and then solve?
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  2. #2
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    Quote Originally Posted by murielx View Post
    find all complex numbers, z, which satisfy the equation z^2 = 6 -8i

    do i begin by writing z^2 - 6 - 8i = 0 and then solve?
    Yes.

    z^2 - 6 - 8i = 0

    z^2 - (6 + 8i) = 0

    z^2 - (\sqrt{6 + 8i})^2 = 0

    (z + \sqrt{6 + 8i})(z - \sqrt{6 + 8i}) = 0


    z = -\sqrt{6 + 8i} or z = \sqrt{6 + 8i}.


    Now the hard part is evaluating these square roots. I'd advise converting to Polars and using DeMoivre's Theorem.
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  3. #3
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    Another way to do this is to write z= x+ yi so that z^2= x^2- y^2+ 2xyi= 6- 8i

    Now solve the two equations x^2- y^2= 6 and 2xy= -8 for real x and y.
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  4. #4
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    I used the way suggested by HallsofIvy and I have become stuck at a certain point....

    x^2 - y^2 = 6

    2xy = -8


    y = - \frac{4}x

    .:.
    x^2 - (-\frac{4}x)^2 = 6

    6 = x^2 + \frac{16}{x^2}


    so x^4 - 6x^2 + 16 = 0


    here is where I am stuck and cant seem to find the two values of x

    I know what to do after finding these values.

    thanks
    Last edited by murielx; March 8th 2010 at 12:04 AM.
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