Math

posted by Majek

Given that the equation x(x-2p)=q(x-p) has real roots for all real values of p and q. If q=3, find a non-zero value for p so that the roots are rational.

  1. Steve

    x(x-2p)=3(x-p)
    x^2-2px = 3x-3p
    x^2-(2p+3) + 3p = 0

    for rational roots, the discriminant must be a perfect square. That is,

    (2p+3)^2-12p
    = 4p^2+12p+9-12p
    = 4p^2+9
    must be a perfect square.
    p=2 is one solution

    check:
    x(x-4) = 3(x-2)
    x^2-7x+6 = 0
    (x-1)(x-6) = 0
    Not only rational, but integers!

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