Rational Roots

Restricting all the way to integer coefficients pins the discriminant test down further: for $ax^2+bx+c=0$ with $a,b,c$ integers, $\Delta=b^2-4ac$ is automatically an integer, so

So an integer-coefficient quadratic has rational roots exactly when its discriminant is a perfect square (and $\Delta\ge0$). This carries over to rational-coefficient equations too: multiplying through by the LCM of the denominators of a rational-coefficient equation gives an integer-coefficient equation with the same roots.

Worked illustrations.

Example 3.11 (no real roots). $2x^2-6x+7=0$: $\Delta=36-56=-20<0$, so both roots are non-real (imaginary).

Example 3.12 (equal roots pin down a parameter). $x^2+2(k+2)x+9k^2=0$ has equal roots exactly when $\Delta=0$: $4(k+2)^2-4(9k^2)=0 \Rightarrow (k+2)^2=(3k)^2 \Rightarrow k=4$ or $k=1$. …