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Thread: Perfect Square

  1. #1
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    Perfect Square

    Determine all integers such that $\displaystyle x^2 + 19x + 92$

    is a square.
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  2. #2
    Senior Member I-Think's Avatar
    Joined
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    Let $\displaystyle =k^2$

    $\displaystyle x^2+19x+92-k^2=0$

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

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

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

    Insert k into the equation to receive your answer

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