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Math Help - Quadratics

  1. #1
    Member Furyan's Avatar
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    Quadratics

    Hello

    I've been looking at a question for some time, but haven't been able to find a way to start. The question is:

    'If x^2 - px - q = 0, where p and q are positive integers, which of the following could not equal x^3?'

    4x + 3, 8x + 5, 8x + 7, 10x + 3, 26x + 5

    I know the equation has one positive and one negative root. I've completed the square and found the discriminant, but I can't find a way into the question. Could someone please give me a pointer.

    Thank you.
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  2. #2
    Super Member Quacky's Avatar
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    Re: Quadratics

    I have a solution, but I'm not sure it's the neatest approach.

    x^2=px+q

    x^3=px^2+qx

    But as x^2=px+q,

    x^3=p(px+q)+qx

    x^3=p^2x+qx+pq

    x^3=x(p^2+q)+pq

    We have these options:
    4x + 3, 8x + 5, 8x + 7, 10x + 3, 26x + 5

    First, I checked 26x+5

    p and q are integers and pq=5
    This means p and q are 1 and 5, although I don't know which letter corresponds to which number.

    Can I get p^2+q=26?
    Yes! If p=5 and q=1! So I can rule this out. Check the others: I found one which didn't work.
    Last edited by Quacky; September 24th 2011 at 07:05 PM.
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  3. #3
    Member Furyan's Avatar
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    Re: Quadratics

    Quacky, thank you for taking the time to reply. I see now how I should have approached the problem.
    Last edited by Furyan; September 25th 2011 at 12:24 AM. Reason: Spelling
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