Suppose the cofficients of a quadratic are rational. How can we use the discriminant to determine if the roots are also rational?

To find out if the roots are rational, you need to check if the discriminant $b^2-4ac$ (which is a rational number $s/t$) is a square of a rational number. We can assume that the fraction $s/t$ is in the lowest terms, i.e., $s$, $t$ are coprime integers. Then you just need to find out if $s$ and $t$ are perfect squares (of integers).

For example $1/2x^2-x+1/4$ has discriminant $1/2$ but $2$ is not a perfect square, so the roots are not rational. On the other hand, $1/2x^2-3/2x+1$ has discriminant $9/4-2=1/4$, where both $1$ and $4$ are perfect squares, so roots are rational.