An experiment with a difficult algebra problem

Instead of working through some medium level AMC 10 problems today, I decided it would be fun to walk through a single really challenging problem with my older son.

The problem we chose was #23 from the 2015 AMC 10b:

Problem 23 from the 2015 AMC 10a

Here’s the problem:

The roots of f(x) = x^2 - ax + 2a are integers.  Find the sum of all values of all possible values of a.

Though this problem is difficult, I choose it because my son is able to understand the solution.  Over the course of about 20 minutes we worked through that solution – slowly.  There were a few misconceptions along the way, but I hope this was a productive exercise.

We began by talking about the problem itself and getting a few initial thoughts from him:

In the second section of the talk he began to try to play around with the values of the roots, but that didn’t go anywhere. Eventually he thinks to try the quadratic formula which leads to an interesting breakthrough:

Now he’s found a new equation that he knows needs to be a perfect square to make the original equation in the problem have integer solutions. Here we start down the path to finding when our new equation is a perfect square, but we hit a strange problem about dividing by zero. This problem causes a little detour:

So, with the division by 0 problem out of the way we returned to trying to find when our new equation could be a perfect square. In this part of the solution he checks a few cases by hand.

Finally, we wrap up by trying to see if we’ve found all of the solutions. There a little bit of number theory / number sense that helps us out here:

So, a tough problem for sure, but nice to see that each individual step is something that my son was able to understand. Learning to pull together all these pieces is an important part of problem solving.