What is the probability that all the roots of x2 + bx + c = 0 are real? [0,1]?

will have real roots when the discriminant of the quadratic, , is non-negative, i.e.

Your question about probability is currently impossible to answer without knowing exactly what the experiment is. Are you picking at random from some interval? Is the choice of either distributed a certain way?

I'll assume the inclusion of "[0,1]" in your question is a suggestion that both are chosen indepently of one another from [0, 1]. Let denote the random variables that take on the values of , respectively. I'll assume are identical and follow the standard uniform distribution, i.e. they each have the same PDF and CDF as below:

where is either of .

Then the question is to find . We have

and we can condition the random variable on the event of by supposing

then integrate over all possible values of .

Bottom left as it's measurements and shape are the closest to the one Jan has