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Math (Quadratics)(Discriminants)

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If a is a real number and a cubic equation
f(x)=x^3−8x^2+(a+12)x−2a
has a repeated root, what is the sum of all the possible values of a?

  • Math (Quadratics)(Discriminants) - ,

    The repeated root cannot be complex, nor can the other.

    If f(x) = (x-h)^2 (x-k) then

    x^3 - (2h+k)x^2 + (2hk+h^2)x - kh^2 = x^3-8x^2+(a+12)x-2a

    so

    2h+k = 8
    h(2k+h) = a+12
    kh^2 = 2a

    k = 8-2h
    h(16-3h) = a+12
    h^2(4-h) = a

    h(16-3h) = h^2(4-h) + 12
    h^3 - 7h^2 + 16h - 12 = 0
    (h-2)^2 (h-3) = 0

    So,
    h = 2 or 3
    k = 4 or 2
    a = 8 or 9

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