Suppose the cofficients of a quadratic are rational. How can we use the discriminant to determine if the roots are also rational?
Firstly the discriminant has to be nonnegative.
I've had a read of Rational root theorem. I kinda get it but i'm not quite sure how to apply it to find a solution for the problem.
$$\sqrt{b^2-4ac} = \frac{p}{q}$$, where $p, q \in \mathbb{Z} $ and are coprime. $$b^2-4ac = \frac{p^2}{q^2}$$ $$ q^2(b^2-4ac) = p^2$$ Am I going in the right direction ? I want reason that some $p/q$ does exist. Please guide me.