The Problem


        f(x)  =  ax2 + bx + c

If a,b, and c are chosen randomly from the interval [0,1], what is the probability that f has real roots?


The Solution

This is equivalent to finding the volume of the solid that lies inside the unit cube that lies above the discriminate surface

        z2  -  4xy  =  0

(Here z is b, x is a, and y is c.)


The thing to notice is that the outer limits of the triple integral is not the unit square since the surface rises above  z  =  1 for part of the square.  This mistake will lead to the answer of 1/9.  Instead it is the part of the unit square that does not lie above the curve

        4xy  =  1

which is shown below


We will need to break this up into two integrals as follows


The solution is approximately equal to .25 which is significantly greater than 1/9.

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