This question comes from Professor Wright via Twitter:

Choose integer coefficients a, b, and c randomly, each between 1 and 100. How often can ax^2+bx+c be factored?

— Prof. Wright (@mathtweep) September 12, 2013

When Professor Wright means factor, he means, be written in the form (dx + e)(fx + g) with d, e, f, and g all being rational numbers.

Here are some questions to extend this investigation:

How often does ax^{2}+bx+c factor with rational values of a, b, and c? Does the range of numbers we choose to look at matter? What if we allow d, e, f, and g to be irrational numbers? How often does this factor if a is 1? What if a is 2? Is there a relationship between the value of a, and the likelihood the quadratic function factors? Can you write a computer program to test out your hypothesis?