Probability of $ax^2 + bx + c = 0$ having real solutions 
$a$, $b$, $c$ are random integer numbers between $1$ and $100$ (including $1$ and $100$, and uniformly distributed). What is the probability that the equation $ax^2 + bx + c = 0$ has real solutions?

This is from a final high school math exam, and I don't know how to get the answer without using computer.

Interesting question.  What kind of class was this?
We can approximate this in the continuous case by letting $(a, b, c)$ reside in the open unit cube in the positive octant.  Given any value of $b$ in the interval $(0, 1)$, the allowable values of $(a, c)$ are those in the region of the first-quadrant unit square "inside" the curve $4ac = b^2$.  The area of this region is
$$
\frac{b^2}{4} + 2\int_{a=b/2}^1 \frac{b^2}{4a} \, da
    = \frac{b^2}{4}-\frac{b^2}{2}\ln\frac{b}{2}
$$
We can then integrate this area over $b \in (0, 1)$:
$$
P = \int_{b=0}^1 \left( \frac{b^2}{4}-\frac{b^2}{2} \ln \frac{b}{2} \right)\,db
  = \left. \frac{5b^3}{36}-\frac{b^3}{6}\ln\frac{b}{2} \right]_{b=0}^1
  = \frac{5+6\ln 2}{36} \doteq 0.25441
$$
But I'm not sure how you get this in the discrete case without some extensive counting.
