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Math Help - Algebra modulus problem

  1. #1
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    Algebra modulus problem

    I am having problems with the following :

    Z +|Z|^2 = 10-2i (the solution must be in the form a+bi)

    I have gotten to :

    a +bi + a^2 +b^2 = 10 -2i

    and cannot think how to solve it

    Any help would be appreciated.

    Jeff
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  2. #2
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    Re: Algebra modulus problem

    You don't know how to compare the real and imaginary parts of the two sides?
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  3. #3
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    Re: Algebra modulus problem

    Ok solvedit. a^2 and b^2 are real numbers. Don't know why I never seen that. You're reply helped me see that. Thank you emakarov
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    Re: Algebra modulus problem

    I'm still having problems with this, I got

    a + bi + a^2 +b^2 = 10 -2i

    but I don't know how to proceed. Apparently you solved it with the help of previous post, but I don't see how. A little hint, please?
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  5. #5
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    Re: Algebra modulus problem

    Quote Originally Posted by Vistus View Post
    I'm still having problems with this, I got

    a + bi + a^2 +b^2 = 10 -2i

    but I don't know how to proceed. Apparently you solved it with the help of previous post, but I don't see how. A little hint, please?
    As you were told, compare the real and imaginary parts.

    You have \displaystyle \begin{align*} a + a^2 + b^2 + b\,i = 10 - 2i \end{align*}, so \displaystyle \begin{align*} a + a^2 + b^2 = 10 \end{align*} and \displaystyle \begin{align*} b = -2 \end{align*}. Solve for \displaystyle \begin{align*} a \end{align*}.
    Thanks from Vistus
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    Re: Algebra modulus problem

    With other words a+a^2+4=10
    a+a^2=6
    a=???

    If i did not get it why b=-2 its because bi=-2i
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  7. #7
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    Re: Algebra modulus problem

    Quote Originally Posted by Petrus View Post
    With other words a+a^2+4=10
    a+a^2=6
    a=???

    If i did not get it why b=-2 its because bi=-2i
    Surely if bi = -2i, then dividing by i gives b = -2.

    Also, if you're studying complex numbers, surely you know how to solve a quadratic equation (such as factorising and using NFL, quadratic formula, completing the square...)
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  8. #8
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    Re: Algebra modulus problem

    Quote Originally Posted by Prove It View Post
    Surely if bi = -2i, then dividing by i gives b = -2.

    Also, if you're studying complex numbers, surely you know how to solve a quadratic equation (such as factorising and using NFL, quadratic formula, completing the square...)
    Ehmm are u saying that i did it wrong? U confused me
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  9. #9
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    Re: Algebra modulus problem

    Quote Originally Posted by Petrus View Post
    Ehmm are u saying that i did it wrong? U confused me
    You said you didn't understand why b = -2. I was explaining why.
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  10. #10
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    Re: Algebra modulus problem

    Quote Originally Posted by Vistus View Post
    I'm still having problems with this, I got

    a + bi + a^2 +b^2 = 10 -2i

    but I don't know how to proceed. Apparently you solved it with the help of previous post, but I don't see how. A little hint, please?
    two complex numbers:

    a+bi and c+di are equal if and only if: a = c, and b = d.

    so if a2+b2+ a + bi = 10 - 2i, then:

    a2 + b2 + a = 10, and b = -2.

    since b = -2, a2 + b2 + a = a2 + 4 + a.

    since we know this already equals 10, we get:

    a2 + a - 6 = 0

    and a2 + a - 6 = (a - 2)(a + 3), so a = 2, or -3.

    as a final check, we verify that:

    z = 2 - 2i
    z = -3 - 2i both satisfy:

    z + |z|2 = 10 - 2i

    2 - 2i + |2 - 2i|2 = 2 - 2i + 4 + 4 = (2 + 4 + 4) - 2i = 10 - 2i

    -3 - 2i + |-3 - 2i|2 = -3 - 2i + 9 + 4 = (-3 + 9 + 4) - 2i = 10 - 2i (checked).
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  11. #11
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    Re: Algebra modulus problem

    Quote Originally Posted by Prove It View Post
    You said you didn't understand why b = -2. I was explaining why.
    Ohh no i did understand was just Teling him if he did not get it sometimes My iPhone Change when i say "u" to "i" i was just making it easy to understand but maybe waste, My fault
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