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Math Help - Complex Number finding the roots

  1. #1
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    Complex Number finding the roots

    Hi

    I need help on the following equation:

    1) z^4+2z^2+4 =0

    u=z^2
    u^2+2u+4=0
    use quadratic equation:

    i get

    u= \frac{-2 +\sqrt{16}}{4}

    or

    u= \frac{-2 -\sqrt{16}}{4}

    my question is do i simplify this to find the roots or leave it?

    P.S
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  2. #2
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    I think completing the square might be better in this case...

    \displaystyle z^4 + 2z^2 + 4 = 0

    \displaystyle z^4 + 2z^2 + 1^2 - 1^2 + 4 = 0

    \displaystyle (z^2 + 1)^2 + 3 = 0

    \displaystyle (z^2 + 1)^2 = -3

    \displaystyle z^2 + 1 = \pm \sqrt{3} i

    \displaystyle z^2 = -1 \pm \sqrt{3} i


    From here, you should convert to polars to find the four values of \displaystyle z.
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  3. #3
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    Quote Originally Posted by Paymemoney View Post
    Hi

    I need help on the following equation:

    1) z^4+2z^2+4 =0

    u=z^2
    u^2+2u+4=0
    use quadratic equation:

    i get

    u= \frac{-2 +\sqrt{16}}{4}

    or

    u= \frac{-2 -\sqrt{16}}{4}

    my question is do i simplify this to find the roots or leave it?

    P.S
    You are aware that \sqrt{16}= 4 aren't you? I can't imagine not using that simplification!
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  4. #4
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    Hello, Paymemoney!

    Your arithmetic is off . . .


    1) Solve: . z^4+2z^2+4 \:=\:0

    Let: u=z^2

    Substitute: u^2+2u+4\:=\:0

    Use Quadratic Formula: . u= \dfrac{-2 +\sqrt{16}}{4} .no

    Did you forget the Quadratic Formula?
    For ax^2 + bx + c \:=\:0 the roots are: . x \;=\;\dfrac{-b \pm\sqrt{b^2-4c}}{2a}

    \text{We have: }\:u^2 + 2u + 4 \:=\:0 \quad\Rightarrow\quad \begin{Bmatrix}a\:=\:1\\b\:=\:2\\c\:=\:4 \end{Bmatrix}

    \text{Hence: }\;u \;=\;\dfrac{\text{-}2 \pm\sqrt{2^2-4(1)(4)}}{2(1)} \;=\;\dfrac{\text{-}2\pm\sqrt{\text{-}12}}{2}

    . . . . . . u \;=\;\dfrac{\text{-}2\pm2\sqrt{\text{-}3}}{2}  \;=\;\text{-}1 \pm i\sqrt{3}


    Back-substitute: . z^2 \;=\;\text{-}1\pm i\sqrt{3}


    . . Therefore: . z \;=\;\pm\sqrt{\text{-}1 \pm i\sqrt{3}}

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  5. #5
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    oops yeh my bad, my quadratic equation was the mistake
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  6. #6
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    Just a quick question if i had something like this how would i find the roots

    z^4+2z^3+5z^2+8z+4=0
    This is what i have done

    u=z

    u^3+12u^2+5u+8u+4=0
    u^3+12u^2+13u+4=0

    what would you do next?
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  7. #7
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    Quote Originally Posted by Paymemoney View Post
    Hi

    I need help on the following equation:

    1) z^4+2z^2+4 =0

    u=z^2
    u^2+2u+4=0
    use quadratic equation:

    i get

    u= \frac{-2 +\sqrt{16}}{4}

    or

    u= \frac{-2 -\sqrt{16}}{4}

    my question is do i simplify this to find the roots or leave it?

    P.S
    Your solutions for u are wrong.
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  8. #8
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    Quote Originally Posted by Paymemoney View Post
    Just a quick question if i had something like this how would i find the roots

    z^4+2z^3+5z^2+8z+4=0
    This is what i have done

    u=z Mr F says: What does this achieve?

    u^3+12u^2+5u+8u+4=0 Mr F says: This is totally wrong.
    u^3+12u^2+13u+4=0

    what would you do next?
    It's very clear that z = -1 is a solution to z^4+2z^3+5z^2+8z+4=0. Therefore z + 1 is a factor. Divide z + 1 into z^4+2z^3+5z^2+8z+4 to get a cubic factor. It is very clear that z + 1 is also a factor of the cubic. Divide it into the cubic to get a quadratic factor. Factorise the quadratic in the usual way.

    From the factor you get the solutions. There is sure to be examples like this in your class notes and textbook. Go back and review this material.
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