Problem Similar SMO(S) 2009, Find probability complex roots k,m in (0,2), x^2 -root(k)x + m = 0

Fascinating problem similar to one from the 2009 Singapore Summer Math Olympiad dealing with the probability of finding complex zeros if two of the coefficients of quadratic equation are randomly selected from real number interval (0,2). The discriminant played significant role in the solution along with the area interpretation of probability. A scaled graph was sketched where the ratio of the part and the whole was the sought after probability.

Source: A Potpourri of Algebra, March 27, 2012
Khor Shi-Jie