Question

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.

Answer 1

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!