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Given that the roots of 2x^2-4x+5 are alpha and beta,find the value of ...
(i)alpha^3beta^3
(ii)(alpha-beta)^2

  • maths -

    Using the sum and product of roots property, we know
    a+b = - (-4/2) = 2 and ab = 5/2, where a and b are alpha and beta

    we know that
    (a+b)^3 = a^3 + b^3 + 3a^2b + 3ab^2
    (a+b)^3 = a^3 + b^3 + 3ab(b+a)
    2^3 = a^3 + b^3 + 3(5/2)(2)
    8 = a^3 + b^3 + 15
    a^3 + b^3 = -7

    in the same way....
    (a-b)^2 = a^2 - 2ab + b^2
    = a^2 +b^2 - 2ab

    but a^2 + b^2 = (a+b)^2 - 2ab

    so
    (a=b)^2 = (a+b)^2 - 2ab - 2ab
    = 4 - 5/2 - 5/2
    = -1

  • silly typo - maths -

    in the third last line I should have had

    (a-b)^2 = (a+b)^2 - 2ab - 2ab

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