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Math Help - A olympiad question

  1. #1
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    Post A olympiad question

    If p and q are the roots of the equation 3x^2 + x- 1 = 0 then prove that 3(p^3 + q^3) + (p^2 +q^2) - (p + q) = 0
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  2. #2
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    Quote Originally Posted by anshulbshah View Post
    If p and q are the roots of the equation 3x^2 + x- 1 = 0 then prove that 3(p^3 + q^3) + (p^2 +q^2) - (p + q) = 0

    HI

    3x^2+x-1=0

    x^2+\frac{1}{3}x-\frac{1}{3}=0

    p+q=-\frac{1}{3} , pq=-\frac{1}{3}

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

    =(p+q)[(p+q)^2-3pq]

    =(-\frac{1}{3})[(-\frac{1}{3})^2-3(-\frac{1}{3})]

    Then

    p^2+q^2=(p+q)^2-2pq

    =(-\frac{1}{3})^2-2(-\frac{1}{3})

    Then putting them together would give you 0 .
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