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Math Help - Quadratics #1

  1. #1
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    Exclamation Quadratics #1

    If the roots of (c-a)x^2 + (a-b)x + (b-c) = 0 are equal, show that a,c,b are in arithmetic progression.

    What I did was:

    For the roots to be equal, the disciminant must be equal to zero.
    Thus, discriminant = (a-b)^2 - 4(c-a)(b-c) = 0

    Eventually, I got a^2 + 6ab + b^2 + 4c^2 - 4bc - 4ac = 0 but I don't know how to use this to solve the problem.

    Please, any help?
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  2. #2
    MHF Contributor red_dog's Avatar
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    You made a mistake. The discriminant is
    a^2+b^2+4c^2+2ab-4bc-4ac=(a+b-2c)^2
    The discriminant must be 0, so a+b-2c=0\Rightarrow c=\frac{a+b}{2}, then a, c, b are in arithmetic progression.
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