# Moving beyond the quadratic formula

It has been interesting watching my son learn a bit more algebra this year. He seems to have made quite a bit of progress studying linear equations, but he’s still at a point where the quadratic formula is the first thing he thinks about with any sort of non-linear equations (which, given that he’s just learning algebra are mostly quadratic equations).

Today he ran across a problem that was probably designed quite specifically to help kids see beyond the quadratic formula. The problem is #20 from the 2007 AMC 10 a

Here’s the problem:

Suppose that the number \$a\$ satisfies the equation $4$ = $a + a^{ - 1}$. What is the value of $a^{4} + a^{ - 4}$?

This problem gave him some difficulty and I asked him to explain his original approach using the quadratic formula first:

Next we talked about how to approach this problem without solving for $a$ first. We had briefly talked through this approach in the morning but this was his first time trying to explain it.

Next we went to Wolfram Alpha to see that the solution he’d found with the quadratic formula actually produced the answer of 194. After that we talked about the graph of $y = x + 1/x$. It was a little hard for him to see that the minimum value on the graph occurred at $x = 1$, but zooming in a little helped him see it.

Finally, I wrapped up by showing him one way that we could use the quadratic formula to help us see where that minimum occurred. I took this approach to help him see that even though the quadratic formula wasn’t so helpful in solving the original problem, it still could be helpful as a way to learn a little bit about x + 1/x.

So, a nice little problem that provides a good example of a situation where the quadratic formula isn’t so helpful. Hopefully examples like this one will help him see that there are lots of to approach non-linear equations, and the quadratic formula is just one of them.