In 1998, I took a multivariable calculus class at my local community college as a high school senior. The professor was Henri Feiner. He had a thick, eastern European accent, and —if I remember correctly— a very dry sense of humor.
One evening, he was working on a problem on the board, and needed to use the quadratic formula. He applied it and started to go on with the problem. But then he stopped and asked, “No one has any questions about what I just did?”. It had seemed a bit funny: It was definitely the quadratic formula or something like it. Most of the numbers were in the right place, but not all.
He then showed us what he had done. He'd used a simplification that applies when the middle coefficient is even. Indeed, if we have:
ax² + 2bx + c = 0 ,
x = [ -2b ± √(4b² – 4ac) ] / 2a .
But now a factor of 2 cancels from numerator and denominator to yield:
x = [ -b ± √(b² – ac) ] / a .
Essentially this is the quadratic formula where you take half the middle coefficient, and then use the quadratic formula "without the numbers": just ac instead of 4ac , and just a instead of 2a .
It is a very helpful simplification. For example, if x² + 14x - 17 = 0 , then we can immediately write down x = -7 ± √(49 + 17) , instead of x = [-14 ± √(14² + 68)] / 2 . Even if we know 14² = 196 , we still have to work out the radicand 164 , only to discover that a factor of 4 is lurking, factor it out and then simplify. But why bother if we know such a simplification can always be performed?
And it’s not like this is some esoteric case. As Professor Feiner pointed out, the middle coefficient will be even half the time, so this simplification will help with half the quadratics you have to solve…
Or does it? I have repeated that spiel for nearly 25 years, but tonight I started thinking about it more carefully. The middle coefficient can indeed be expected to be even half the time, but the simplification will not help you with half the quadratics.
When I realized this, I looked up Professor Feiner's phone number and called him at home. He is 86 years old, just retired. He was intrigued to hear the answer and we launched immediately into a discussion. (He asked, "Were you the kid who corrected me on a series problem once?". I said that sounded like it could have been me.)
So the question is: What percentage of the time will this simplification actually be helpful?