Q:

(simple) find value of m which makes a quadratic function continuous on R, please add explanation for the discriminant usage

I need explanation on the usage of the discriminant  $b^2 - 4ac$ to answer this question please.
If $$f(x) = \frac{(x+3)}{(x^2 + mx +9)}$$ is continuous on $\Re$, then $m$  $\epsilon$ ........

A:

Start with $g(x)=x^2+mx+9$ for $x\in\Bbb R$, then look for the zeroes of $g$:
$$x={-m\pm\sqrt{m^2-36}\over 2}$$
If $|m|\ge 6$ then the zeroes of $g$ are real numbers.  If not, then $g$ has no real zeroes.  In particular, the discriminant, the value $b^2-4ac = m^2-36$ is the deciding factor in this problem, since we take the square root of $m^2-36$.  If $m^2-36$ is negative, i.e. $m^2 \lt 36$, the square root introduces non-real solutions, which is another way of saying that $g(x)$ never crosses the zero line ($g(x)=0$).  Can you continue from here?

