## Saturday, August 22, 2015

So ... I've seen a couple of YouTube videos that feature songs about the Quadratic Formula.  I often see it written like this:

$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

and it occurs to me that I've always written it this way:

$x = \dfrac{-b}{2a} \pm \dfrac{\sqrt{b^2-4ac}}{2a}$

Which one is better?

1. Our standard answer: it depends. In solving the following quadratic equation exercise, it's not immediately clear which format would be "better" (or maybe neither is useful):

2. I imagine either can be of use. I've always used the top one.

3. Better for what purpose? They'll get you the same answer. The top one is easier to remember; the bottom one will get you to the final conventional form faster when the root is irrational. So for teaching, I'd say the top one is usually better, but for mathematical practice the bottom one is probably better.

4. personally, i think ±sqrt(-c/a+(b/(2a))^2)-b/(2a) is best. (Historical/geometric completing the square).

I don't really like either of the two above, but of the two above, I prefer the bottom one, because it's very slightly easier to use it to talk about the location of the vertex being the arithmetic mean of the roots.

My students, though, hate it when I use the bottom one. They're all used to the top one. I think that says something about how they've learned algebraic equivalence, but I'm not going to dwell on it here.

5. The bottom one emphasizes the line of symmetry (first term) and demonstrates the symmetric nature of the parabola at any pair of solutions. Nonetheless, I teach them to sing the top one.

6. It's not a matter of one form being "better" or "worse" than the other; it is more a matter of which form is appropriate for the current application. When writing with pen and paper, and depending upon my audience, I generally take the equation as far as the first form. It is more concise. However, because so much of my own work involves computer programming, and needs to include the option for complex solutions, the second form often gets applied. In any situation for which the solutions cannot be assumed to be real numbers, the two terms are going to be dealt with separately, so you end using the second form. Might as well start with it in these situations.