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Math Help - Complex Number

  1. #1
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    Complex Number

    Have a question:

    Let f be the complex polynomial
    f(x)=(3+i)x^2+(-2-6i)x+12
    Calculate the following values f(i) and f(2+3i)
    I calculated these values as such:
    f(i)=(3+i)i^2+(-2-6i)i+12
    Doing the math f(i)=15-3i
    f(2+3i)=(3+i)(2+3i)(2+3i)+(-2-6i)(2+3i)+12
    Doing the math f(2+3i)=-1+13i

    Is this right?

    Also, it then asks find a complex number z such that the equation f(x)=z has a unique solution (use the formula for solving a quadratic equation). How do you do this?
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  2. #2
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    Hello, JaysFan31!

    Let f be the complex polynomial: f(x) .= .(3 + i)x + (-2 - 6i)x + 12
    Calculate the following values: f(i) and f(2 + 3i)

    I calculated these values as such:

    f(i) .= .(3 + i)i^2 + (-2 - 6i)i + 12
    . . Doing the math: .f(i) .= .15 - 3i

    f(2 + 3i) .= .(3 + i)(2 + 3i)(2 + 3i) + (-2 - 6i)(2 + 3i) + 12
    . . Doing the math: .f(2 + 3i) .= .-1 + 13i

    Is this right? . . . . They look good to me!


    Also, it then asks find a complex number z
    such that the equation f(x) = z has a unique solution.
    (Use the Quadratic Formula. .How do you do this?

    We have: .(3 + i)x + (-2 - 6i)x + 12 .= .z

    Or: . (3 + i)x + (-2 -6i)x + 12 = z .= .0

    We have a quadratic equation with: a = (3 + i), b = (-2 - 6i), c = (12 - z)
    . . . . . . . . . . . . . . . . . . . . . . . . . _____________________
    . . . . . . . . . . . . . . . . -(-2 - 6i) √(-2 - 6i) - 4(3 + i)(12 - z)
    Then we have: . x .= . ----------------------------------------------
    . . . . . . . . . . . . . . . . . . . . . . . . . . 2(3 + i)

    A quadratic will have one solution if its discriminant equals zero.
    That is: . (-2 - 6i) - 4(3 + i)(12 - z) .= .0

    Now solve for z . . .

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  3. #3
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    Thanks a lot. You're very helpful. Where did you teach?
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  4. #4
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    Hello again, JaysFan31!

    I taught at a tiny two-year state college,
    . . Massachusetts Bay Community College, in Wellesley, MA.

    I enjoyed it so much, I taught there for 36 years.

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