Open-ended problem: How likely is it a quadratic equation will factor?

This question comes from Professor Wright via Twitter:

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 ax2+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?

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5 thoughts on “Open-ended problem: How likely is it a quadratic equation will factor?

    • 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?).

  1. 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.

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