How to use the discriminant to find out what type of solutions the equation has for qx^2+rx+s=0?

The discriminant of $q {x}^{2} + r x + s = 0$ is given by the formula:
$\Delta = {r}^{2} - 4 q s$
If $\Delta < 0$ then the quadratic equation has no real solutions. It has two distinct complex roots (complex conjugates of one another).
If $\Delta = 0$ then the quadratic equation has one repeated root. If $q$, $r$ and $s$ are rational then the repeated root is rational too.
If $\Delta > 0$ then the quadratic equation has two distinct real roots. If $q$, $r$ and $s$ are rational and $\Delta$ is the square of a rational number, then the roots will be rational too.