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Math Help - Help with complex numbers

  1. #1
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    Help with complex numbers

    Hi please see the attached pdf. I need help with the following

    10.) B

    11.)

    12

    13.

    I'm confused. I already know all the answers because they are given. What i need is an explanation/guidance.
    Thanks
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Help with complex numbers

    10.)

    b) \frac{1}{2-i}-\frac{1}{2+i}=\frac{(2+i)-(2-i)}{2^2-i^2}=\frac{2i}{5}=0+\frac{2}{5}i

    11.) \frac{x}{1-i}+\frac{y}{1+3i}=2

    \frac{x(1+3i)+y(1-i)}{(1-i)(1+3i)}=2

    \frac{x+3xi+y-yi}{1+2i-3i^2}=2

    \frac{(x+y)+(3x-y)i}{4+2i}=2

    (x+y)+(3x-y)i=8+4i

    Equating coefficients, we obtain the system:

    x+y=8

    3x-y=4

    Adding, we find:

    4x=12

    x=3 and so

    y=5

    12.) (z+i)(2-i)=3-4i

    z+i=\frac{3-4i}{2-i}\cdot\frac{2+i}{2+i}=\frac{10-5i}{5}=2-i

    z=2-2i

    13.) k(x-(2+3i))(x-(2-3i))=k((x-2)-3i)((x-2)+3i)=

    k\left((x-2)^2-(3i)^2 \right)=k\left(x^2-4x+4+9 \right)=k\left(x^2-4x+13 \right)=0

    where 0\ne k\in\mathbb{R}
    Thanks from Benja303
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