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?

Doesn’t it make more sense to restrict d,e,f and g to integers, since that’s what a,b and c are?

That’s basically the gist of the original question posed on Twitter, so yes, that is a good question too. It prompted me to write this post!

I think it is also interesting to ask when can we make the coefficients irrationals (or can we make them irrationals?).

The roles a and c play are interchangeable, which is interesting to look at.

David

It seems that I was told once that fewer than 5% of quadratics factor even when you restrict a, b, and c to values between – 10 and 10. In any case, it’s certainly a small percentage.

If you mean “factor with rational coefficients”, then yes, the answer is surprisingly small. If you mean “factor at all”, I believe the answer is about 25%.