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Math Help - Complex quadratic equation problem

  1. #1
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    Difficult quadratic equation problem

    Hi, I have an 'advanced quadratic' problem which I'm not sure how to solve.

    Let pairs (a, c) and (b, d) be roots of the equation x^2 + ax - b = 0 and x^2 + cx + d = 0 respectively. Find all possible real values for a, b, c, d.

    Thanks for any help, BG
    Last edited by BG5965; August 30th 2009 at 08:03 AM.
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  2. #2
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    Here's a start.

    Since (a,c) are the roots of x^{2}+ax-b, then we can write it as

    (x-a)(x-c)=x^{2}+ax-b

    For the other, we can write:

    (x-b)(x-d)=x^{2}+cx+d

    Try equating coefficients and see what you come up with.

    Some advice. I assume when you wrote 'complex' you meant the problem was difficult. Do not use the word complex unless you are referring to the non-real numbers. It may add confusion. Just a thought.
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  3. #3
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    I think we could use Viete's formula:

    x_1 + x_2 = -\frac{b}{a}

    x_1 \times x_2 = \frac{c}{a}

    So in this case, for equation 1, with (a, c) = (x_1, x_2) we could do: x_1 + x_2 = -\frac{b}{a} and x_1 \times x_2 = \frac{c}{a} and with (b, d) being (x_3, x_4), x_3 + x_4 = -\frac{b}{a} and x_3 \times x_4 = \frac{c}{a}

    Just a suggestion, I'm not too sure what to try apart from that.
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