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Math Help - Perfect Square

  1. #1
    Member
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    Perfect Square

    Determine all integers such that x^2 + 19x + 92

    is a square.
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  2. #2
    Senior Member I-Think's Avatar
    Joined
    Apr 2009
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    288


    Let =k^2

    x^2+19x+92-k^2=0

    Apply quadratic formula:  x=\frac{-19\pm{\sqrt{19^2-4(92-k^2)}}}{2}
    x=\frac{-19\pm{\sqrt{4k^2-7}}}{2}

    Determine integers such that 4k^2-7=m^2
    4k^2-m^2=7
    (2k+m)(2k-m)=7
    As k and m are integers, these 2 factors must be factors of 7

    2k+m=7
    2k-m=1
    4k=8\rightarrow{k}=2

    Insert k into the equation to receive your answer

    x=\frac{-19\pm3}{2}
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