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Math Help - Complex numbers

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    Complex numbers

    Determine all complex solutions of the equation
    z^4 - 4z^2 + 16 = 0
    in Cartesian form.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by mamt6 View Post
    Determine all complex solutions of the equation
    z^4 - 4z^2 + 16 = 0
    in Cartesian form.
    Note that this equation is quadratic in z^2.

    Take w=z^2 to turn the equation into w^2-4w+16=0.

    Apply the quadratic formula to get w=\frac{4\pm\sqrt{16-64}}{2}=\frac{4\pm4\sqrt{3}i}{2}=4\left(\tfrac{1}{  2}\pm\tfrac{\sqrt{3}}{2}i\right).

    Note that in complex polar form we have w_1=4\left[\cos\!\left(\tfrac{\pi}{3}\right)+i\sin\!\left(\tf  rac{\pi}{3}\right)\right] and w_2=4\left[\cos\!\left(\tfrac{5\pi}{3}\right)+i\sin\!\left(\t  frac{5\pi}{3}\right)\right]

    To get the four solutions for z, evaluate z_{1,2}=w_1^{1/2} and z_{3,4}=w_2^{1/2} by applying DeMoivre's Theorem.

    Can you finish this off?
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