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Math Help - Solve quadratic into complex numbers

  1. #1
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    Question Solve quadratic into complex numbers

    Hi

    I have this equation:
    s^2 + \sqrt{2}s + 4

    and it somehow factorises to this:

    (s + 2e^{+j(\pi/4)})(s + 2e^{-j(\pi/4)})

    I've tried using the quadratic formula but cant get it into the form with e^{x}

    Would anyone be able to shed a little light on this?

    Thanks
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by eggy524 View Post
    Hi

    I have this equation:
    s^2 + \sqrt{2}s + 4

    and it somehow factorises to this:

    (s + 2e^{+j(\pi/4)})(s + 2e^{-j(\pi/4)})

    I've tried using the quadratic formula but cant get it into the form with e^{x}

    Would anyone be able to shed a little light on this?

    Thanks
    Well it doesn't if you multiply it out you get

    (s + 2e^{+j(\pi/4)})(s + 2e^{-j(\pi/4)})=
    \displaystyle s^2+2se^{-j\frac{\pi}{4}}+2se^{j\frac{\pi}{4}}+4
    \displaystyle s^2+2s\left(e^{j\frac{\pi}{4}}+e^{-j\frac{\pi}{4}} \right)+4
    \displaystyle s^2+2s(2\cos\left(\frac{\pi}{4} \right))+4
    s^2+2\sqrt{2}s+4
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  3. #3
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    Quote Originally Posted by eggy524 View Post
    Hi

    I have this equation:
    s^2 + \sqrt{2}s + 4

    and it somehow factorises to this:

    (s + 2e^{+j(\pi/4)})(s + 2e^{-j(\pi/4)})

    I've tried using the quadratic formula but cant get it into the form with e^{x}

    Would anyone be able to shed a little light on this?

    Thanks
    When you use the quadratic formula to solve the equation you get s=\frac{-\sqrt{2}\pm\sqrt{2- 4(1)(4)}}{2}= \frac{-\sqrt{2}\pm\sqrt{-14}}{2}
    so that x^2+ \sqrt{2}x+ 4= \left(s+ \frac{\sqrt{2}}{2}+i\frac{\sqrt{14}}{2}\right)\lef  t(s+ \frac{\sqrt{2}{2}}- i\frac{\sqrt{14}}{2}\right).

    You can change those numbers to "polar" form but they are not anything like " 2e^{i\pi/4}" or " e^{-i\pi/4}.

    The modulus is correct: \sqrt{\frac{1}{2}+ \frac{7}{2}}= \sqrt{4}= 2
    but the argument is tan^{-1}\left(\frac{\sqrt{14}}{2}\frac{2}{\sqrt{2}}\righ  t)= tan^{-1}\right(\sqrt{7}) which is closer to 1.2 radians than to \pi/4.

    Did you, as TheEmptySet suggests, drop a factor of "2" from the coefficient of x?
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  4. #4
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    Thanks guys, makes more sense now!
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