# Thread: Quadratic questions : rational roots problem

1. ## 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.

2. 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.