smc

The discriminant of the quadratic is $b^2-4c$. Since the quadratic has real roots, $$b^2-4c\ge 0$$ $$b^2\ge 4c$$ If $b=6$, then $c$ can be from $1$ to $6$. If $b=5$, then $c$ can also be from $1$ to $6$. If $b=4$, then $c$ can be from $1$ to $4$. If $b=3$, then $c$ can be $1$ or $2$. If $b=2$, then $c$ can only be $1$. If $b=1$, no values of $c$ in the set would work. Thus, there are a total of $19$ equations that work. The answer is $\boxed{19}$.