If $a$ is uniformly distributed over [-22,31], what is the probability that the roots of the equation are both real:
$$x^2+ax+a+63=0$$
I have solved this in another situation. I know that I need the expression under the square root to be positive. So,
$$0 \le b^2-4ac$$
which in my case is
$$0 \le a^2-4a-252$$
but from here I am having a hard time finding out what I should be doing. I tried graphing it but I am not sure that I was doing it properly.
hint
You just need to figure out what values of $a$ obey $a^2-4a-252>0$ and then determine what fraction of the range $(-22,31)$ that is.