What is the origin of the denominator 2 in the formula for work when stretching a spring?

Answer

You have already come up with an important aspect yourself! Indeed, the force is not constant, but increases with the amount the spring is stretched (or compressed): F = kx

The formula for work W = F . So, as you rightly suspect, you cannot apply L because F is not constant during the displacement. In that case the formula (in one dimension) is given by an integral. That’s a very important math concept that (given your age) you may not have learned yet. With such an integral you can take into account the fact that F is not constant. If you then calculate the work you find 0.5 k L². (L = the length that the spring has been compressed or stretched compared to its resting state). Your physics teacher will certainly be able to explain how such an integral is constructed.

As an approximation of the actual work you would divide the stretching of the spring into a lot of very small successive steps of length dx, steps that are so small that you can assume the force as constant during such a step. If you add all those steps together you find the total work. If all steps are the same size, the first steps will require less labor than the last.

Such an integral is a powerful mathematical tool that makes this possible, moreover, you can use the integral to take those steps even infinitely small. During an infinitesimal step the force is indeed constant because you are actually looking at it in only one place. Of course you have to do an infinite number of such steps. But mathematically it is possible to cover a finite distance L by taking an infinite number of infinitely small steps. Such an integral does that. As mentioned, you will see mathematics in class, and you can certainly take this explanation to your physics teacher, because he certainly knows more than enough about those useful integrals.