Consider the quadratic equation Ax2 Bx C 0 where A B and

Consider the quadratic equation Ax^2 + Bx + C = 0, where A, B and C are independent and have uniform distributions on [-n, n], n > 0. Find the probability that the equation has real roots.

Solution

AX^2+Bx+c =0 has real roots if

B^2-4AC >=0

B^2 is always positive

hence A and C should have opposite signs

If A negative C positive or if C negative A positive

A being negative has prob 0.5 similarly for C

Hence for real roots prob = 0.5(0.5)(1) = 0.25 (As B can be either positive or negative)