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Thread: Solve the equation z^4 + 81 = 0

  1. #1
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    Solve the equation z^4 + 81 = 0

    Solve the equation z^4 + 81 = 0
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by norivea
    Solve the equation z^4 + 81 = 0
    $\displaystyle z^4+81=0$

    $\displaystyle z^4=-81$

    Taking the square root of both sides:
    $\displaystyle z^2 = \pm 9i $

    Again taking the square root of both sides:
    $\displaystyle z = \pm \sqrt{\pm 9i}$ = $\displaystyle \pm 3 \sqrt{\pm i}$

    So what is $\displaystyle \sqrt{\pm i}$?

    To answer this, let's write "i" a bit differently using the exponential form for complex numbers:
    $\displaystyle i = cos(\pi/2)+i \, sin(\pi/2) = e^{i\pi/2}$

    Thus
    $\displaystyle \sqrt{i}=(i)^{1/2} = \left( e^{i\pi/2} \right ) ^{1/2}$ = $\displaystyle \left( e^{i\pi/2*1/2} \right ) = e^{i\pi/4}$ = $\displaystyle cos(\pi/4)+ i \, sin(\pi/4) = \frac{\sqrt2}{2} + i \frac{\sqrt2}{2}$ = $\displaystyle \frac{\sqrt2}{2}(1+i)$

    Similarly:
    $\displaystyle \sqrt{-i}=(-i)^{1/2} = \left( e^{-i\pi/2} \right ) ^{1/2}$ = ... = $\displaystyle \frac{\sqrt2}{2}(1-i)$

    So finally the solution to $\displaystyle z^4+81=0$ is

    $\displaystyle z =\pm 3 \sqrt{\pm i}$ =$\displaystyle \pm 3 \frac{\sqrt2}{2}(1+i)$ and $\displaystyle \pm 3 \frac{\sqrt2}{2}(1-i)$

    -Dan
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  3. #3
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    Hello, norivea!

    Solve the equation: $\displaystyle z^4 + 81 \:= \:0$

    We have: .$\displaystyle z^4\:=\:-81$

    The polar form of $\displaystyle -81$ is: .$\displaystyle 81(\cos\pi + i\sin\pi)$

    Then: .$\displaystyle z\;=\;[81(\cos\pi + i\sin\pi)]^{\frac{1}{4}}$

    Hence: .$\displaystyle z\;=\;3\bigg[\cos\left(\frac{\pi}{4} + \frac{\pi}{2}n\right) + i\sin\left(\frac{\pi}{4} + \frac{\pi}{2}n\right)\bigg]$ for $\displaystyle n = 0,1,2,3$

    Therefore: .$\displaystyle z\;=\;\left\{\frac{3}{\sqrt{2}}(1 + i),\;\frac{3}{\sqrt{2}}(1 - i),\;\frac{3}{\sqrt{2}}(-1 + i),\;\frac{3}{\sqrt{2}}(-1 - i)\right \}$


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    Alternate method


    We have: .$\displaystyle z^4\:=\:-81\quad\Rightarrow\quad z^2 \:=\:\pm 9i\quad\Rightarrow\quad z\:=\:\pm\sqrt{\pm9}\sqrt{i}$

    . . Then: .$\displaystyle z\:=\:\pm3\sqrt{i}$ and $\displaystyle \pm3i\sqrt{i}$


    Now, if you happen to know that: .$\displaystyle \sqrt{i}\:=\:\frac{1 + i}{\sqrt{2}}$

    . . then we have: .$\displaystyle z\:=\:\pm 3\left(\frac{1 + i}{\sqrt{2}}\right)$ and $\displaystyle \pm 3i\left(\frac{1 + i}{\sqrt{2}}\right)$

    which gives us the same fourth roots.

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    Re: Solve the equation z^4 + 81 = 0

    $\displaystyle \begin{aligned} x^4+3^4 & = (x^2+3^2)^2-2\cdot3^2x^2 = (x^2+3^2)^2-(3\sqrt{2}x)^2 = (x^2-3\sqrt{2}x+3^2)(x^2+3\sqrt{2}x+3^2). \end{aligned}$

    From here I trust you know how to solve quadratic equations (use the quadratic formula/complete the square).
    Last edited by TheSaviour; Jan 15th 2013 at 12:20 PM.
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    Forum Admin topsquark's Avatar
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    Re: Solve the equation z^4 + 81 = 0

    Quote Originally Posted by TheSaviour View Post
    $\displaystyle \begin{aligned} x^4+3^4 & = (x^2+3^2)^2-2\cdot3^2x^2 = (x^2+3^2)-(3\sqrt{2}x)^2 = (x^2-3\sqrt{2}x+3^2)(x^2+3\sqrt{2}x+3^2). \end{aligned}$

    From here I trust you know how to solve quadratic equations (use the quadratic formula/complete the square).
    Nice idea. One slight typo: The first term past the second = sign should be squared.

    Ummmm...No problem with you helping out, but you are aware that this thread is over 6 years old?

    -Dan
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    Re: Solve the equation z^4 + 81 = 0

    Quote Originally Posted by topsquark View Post
    Nice idea. One slight typo: The first term past the second = sign should be squared.

    Ummmm...No problem with you helping out, but you are aware that this thread is over 6 years old?

    -Dan
    LOOOOOOL! I honestly had no idea. How on earth did it show up on the main/home pages?

    EDIT: Oh, I think I know why. I was looking Who's Online and someone was reading this! Trap!
    Last edited by TheSaviour; Jan 15th 2013 at 12:19 PM.
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