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Math Help - Quadratic questions : rational roots problem

  1. #1
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    Quadratic questions : rational roots problem

    consider a quadratic equation (a+c-b)x^2 + 2cx + (b+c-a) = 0, where a,b,c are distinct real numbers and a+b-c is not equal to 0. Suppose that both the roots of the equation are rational. Then which must hold true?
    1. a,b,c are rational
    2. c/(a-b) are rational
    3. b/(c-a) are rational
    4. a / (b-c) are rational.
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  2. #2
    MHF Contributor red_dog's Avatar
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    The discriminant is \Delta =4(a-b)^2

    Then the roots are:

    x_1=\frac{-2c-2(a-b)}{2(a+c-b)}=-1

    x_2=\frac{-2c+2(a-b)}{2)a+c-b)}=\frac{a-b-c}{a-b+c}=\frac{1-\frac{c}{a-b}}{1+\frac{c}{a-b}}

    Then \frac{c}{a-b} must be rational.
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