The other day, I had a young friend show me a method of factoring that he favoured. He was told that "nobody" knew the reasoning behind this method; it just worked. I couldn't let that go. There is no mathematical method that just "always works" and no one knows why.
Not having been able to find any explanation on the internet, my friend thought I should post an explanation. I thought about it. I even began to prepare a post. But I couldn't do it. To show him a full explanation would be like giving the answer to a problem at the first sign of struggle. I don't want to rob him of an opportunity to work through some algebraic thinking. Instead, like someone stuck while climbing a wall, I am confident he simply needs a bit of a leg up.
Here then, is my challenge: I will show you my work, along with a couple of questions to think about. Can you reason your way toward a proof that your method of factoring will work for any quadratic expression (and why is it an expression, rather than an equation??) that factors?
First, let me remind you of what happens when we multiply two binomials with a leading coefficient of 1.
Next, let's factor the quadratic expression you came up with the other day.
Why is the 4^2 important? Did each step make sense? Which ones do we skip when we apply the method? Will this method always work? Why?