## You don’t suck at math (Part 2)

In my last post, I promised to give a thorough explanation of the quadratic formula–where it comes from.

Here is the formula again.

Now that you’ve seen it, I encourage you to forget it for a while. As I’ve thought about this more, and talked to a few people, I realize that my promise is itself somewhat against the spirit of my last post. That is, I am not convinced that I will be able to give a short explanation of the quadratic formula, without going on a few big tangents–whenever you start talking about math, you realize that there is a lot more going into what you are saying, a lot more ideas that are interacting with the one you are trying to convey, then you thought at first. I consider these tangents, however, to be of much value in themselves. It is often in fudging these extra parts, in order to press on towards your goal, that you are in danger of losing the audience. So I can’t promise to even get all the way to the quadratic formula in this post, though I do hope to say something interesting, and I will get to the quadratic formula in a future post.

We’re going to first talk a bit about equations. First of all, what is an equation, and what does it mean to “solve” it? Well, let’s consider an example. Here is an equation:

Solving this means finding a number that, when substituted for x, makes the equation true. If you don’t know how to solve this, that’s nothing to be embarrassed about. And I’m about to go through it anyway. But I want to stress what it is that we are doing. We want to find a number that, when we multiply it by 3, and then subtract 9 from that, we get zero. I could give real world examples of why you might want to do this, but I’m not going to. Doing so conveys the impression that it is only through such applications that math becomes interesting, whereas nearly the opposite is true: if you think that math is only studied for the sake of a series of contrived applications, you are going to find it very boring indeed.

So, solving this equation is pretty simple–we move the -9 over to the other side, making it positive (“add nine to both sides” if you like to think of it in that way):

and then we divide by 3:

We could solve other such equations in a similar manner:

Answer:

Answer:

Notice that all of the equations I have given so far have 0 on the right hand side. There is a reason I did this. Basically, every equation is like this, if you think about it. For example, if I had the equation:

I could turn it into an equation with zero on the right hand side by subtracting 24 from both sides:

I could do the same thing with even much more complicated equations, by moving *everything* over to one side. Here this would actually be an unnecessary step if you were trying to solve the equation for x, but that’s not the point right now. The point is that, in math, whenever two things are essentially the same, we prefer to consider only one of them, and we like to be fairly consistent in that choice, unless it is convenient to switch (which it often will be, so it is important to know how to go back and forth). So the study of solving equations reduces to the study of how to make any particular expression (which is just one side of an equation) equal to zero. This simplifies things, in a way: instead of continually asking, “how to we make ‘blah’ equal to ‘blech’?” we only have to ask “how do we make ‘blah’ become zero?” This is somewhat analogous to the situation where two people are looking for each other–the task often seems simpler if one of them is asked to stand completely still.

Anyway, for the sorts of equations we’ve looked at so far, we can answer the question of how to solve them *in general*. By that I mean, we can answer it regardless of the different numbers that are showing up in these different equations I’ve given–or, more precisely, we can answer it *in terms* of those numbers. Let’s consider the “general form” for the sorts of equations we’ve looked at:

Here, a and b are thought of as “constants” and x is thought of as the “variable.” When I was first learning algebra, this distinction, I remember, would often enrage me. I had a valid reason: we don’t know what a, x, or b are, so what is the sense in drawing this distinction between two different types of unknowns? Philosophically, it is hard to say–and I won’t try to justify it completely. But I will say that much of the clever work that can be done in algebra is, in some sense, built on taking this distinction seriously, by continually shifting our perspective on what the constants are, and what the variables are.

OK, back to work. How do we solve for x? Well, in the same way we have been solving. First let’s move the b over:

Next, we divide by a:

And we are done.

The above was really the simplest sort of equation to be solved. I’m sure you’ve seen more complicated equations; for example, here is one:

To solve this would be to find values for x and y that make the equation true. So here we’d be looking for a pair of numbers–and there may be many such pairs. One such pair is . Another is . In fact, there are infinitely many pairs that work. This might seem like a bad thing: who wants a problem that has infinitely many answers? If this bothers you, you can think of all the solutions together as a “set”–then there is only one set of pairs that precisely includes all of the solutions to this equation, and nothing extra. This is a good way of thinking about these things, because often there is a neat geometric description to the set that you are thinking about, allowing us to make a connection between algebra and geometry. It turns out here that the solutions to this equation describe a circle (maybe I’ll explain why, in a future post).

The question I want to ask now is: what made this equation more complicated than the ones before? Well, in our original equations, there was only one variable, x, and it appeared appeared only once, and there wasn’t any exponent on it. Here, however, we have multiple variables (two), and exponents on each of them. Say, now, we wanted to make the “next most complicated” equation to the ones we looked at before. It appears that there are two directions we could go in: we could include more variables, or we could include exponents on the variables that are already there. Doing the former, subject to some further restrictions, leads to the beautiful subject known as linear algebra. To get to the quadratic equation, however, we do the latter: we allow an exponent of 2 on the variable x.

To do that, let us start out with the simplest case of such an equation. Let consider this equation:

Now, I know this isn’t in the form I mentioned before, where zero is on the left. It easily could be (just move the nine over to the right hand side), but, right now, that’s not what I want to focus on. The important point here is that many of us probably already know how to solve this equation: set x equal to 3. Since 3 times 3 is nine, this is a solution. Less obviously, there is another solution, if we are considering negative numbers: -3 times -3 is also 9. This can seem like a pedantic point, but it actually represents an important fact that will also be true of the more difficult quadratic equations: typically, they will have two solutions). So when we allow a power of two on the x, there are (except in some special cases that we will be able to describe) two solutions. Isn’t that nice?

Incidentally, many people are bothered by the fact that -3 times -3 is equal to 9. In my next post, I will give an explanation for this, as I also give a general explanation for how you solve all of the equations of the form I just gave. I mean equations of the form:

where can be anything. These are a special case of quadratic equations, and we’ll have to understand them before we can even hope to get to the more general situation!