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Thread: Solving quadratic equations with Complex numbers

  1. #1
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    Solving quadratic equations with Complex numbers

    Hey guys,
    The problem in question here is- "Solve this equation over C: Z2+Z+(1-i) "

    I have tried completing the square, and ended up with: (Z+1/2)2 - (-3+4i)/4 = 0
    This doesn't seem very helpful, because when I go to factorise, I end up with sqrt(-3+4i)/2, which seems unable to be simplified.

    Using the quadratic equation yields the same result :/

    The solution answers are: Z=-i and Z=-1-i

    Any help much appreciated, thank-you
    Smeato
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  2. #2
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    Re: Solving quadratic equations with Complex numbers

    You need to learn a little more about complex numbers. The complex number (-3+ 4i)/4 can be written in the "polar form" \frac{5}{4}(cos(53.13)- i sin(53.13)). The square roots are \frac{\sqrt{5}}{2}(cos(53.13/2)- i sin(53.13/2))= \frac{\sqrt{t}}{2}(cos(26.575)+ i sin(26.575)) and \frac{\sqrt{5}}{2}(cos(206.575)- i sin(206.575)) where the angles are in degrees.
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  3. #3
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    Re: Solving quadratic equations with Complex numbers

    Why aren't any of my replies showing up in the thread?
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  4. #4
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    Re: Solving quadratic equations with Complex numbers

    $z^2+z+(1-i) = 0$

    $(z^2+1) + (z-i) = 0$

    $(z-i)(z+i) + (z-\color{red}{i}) = 0$

    $(z-i)(z+i+1) = 0$

    $z = i$ or $z = -(1+i)$
    Last edited by skeeter; Mar 10th 2017 at 05:57 PM. Reason: fixed typo
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    Re: Solving quadratic equations with Complex numbers

    Thanks skeeter, but how did you get from line 3 to line 4?
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    Re: Solving quadratic equations with Complex numbers

    Quote Originally Posted by Smeato View Post
    Thanks skeeter, but how did you get from line 3 to line 4?
    line 3 should have $(z-i)$ as the last term ... accidentally hit 1 instead. Following is correct ...

    $\color{red}{(z-i)}(z+i) + \color{red}{(z-i)} = 0$

    pull out the common factor ...

    $\color{red}{(z-i)}[(z+i) + 1] = 0$
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  7. #7
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    Re: Solving quadratic equations with Complex numbers

    Ahhhh, makes much more sense! Thanks a bunch
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