If instead A,B,C ~ uniform(0,1) and are still independent, one way to do it is to compute the distribution of the random variable B^2 - 4AC. The chance of real roots is then the probability that this quantity is positive. I recall that someone on Math Help Forum mentioned recently the distribution of the product of standard uniforms (which you need for the 4AC part), but then you have to find the distribution of the square, and finally the difference too. The distribution would seem to be prety messy. I'm not sure whether folks on this list would deign to use Monte Carlo simulation or numerical methods to do such a calculation, but I found this probability to be about

**0.254** using RAMAS Risk Calc. You can also do it with Monte Carlo simulation in R (

The R Project for Statistical Computing) with a

simple script like

many = 2000000

A = runif(many)

B = runif(many)

C = runif(many)

arg = B^2 - 4*A*C

length(arg[0<arg])/many

Is resorting to such approaches bad form?