Where Does The Quadratic Formula Come From?

If you've ever went through some form of math class, you've likely heard the term linear equation. You'll probably recognize the slope-y-intercept formula y = mx + b. If you completed a whole Algebra I curriculum, you likely recall going beyond these linear equations and studying quadratic equations. They were the ones with an x-squared and looked like this:

All long as a does not equal zero. Look familiar? You might also remember solving these (somewhat ugly-looking) problems. The first way was factoring, and life was easy if you could factor. But what if you couldn't factor? Most of us turned to the universal quadratic formula. You remember that, right? It had that 4ac, the over 2a, the b-squared, the plus or minus... Yes, it was messy, but it worked! But that wasn't the only other way. Some of you, like me, preferred the method called completing the square. Whichever you preferred, the order of learning usually went like this:

1) Factoring

2) Completing the Square

3) Quadratic Formula

Now, I understood learning factoring before the quadratic formula, but I never really knew why I needed to know how to complete the square. Yes, it becomes useful when you study integral calculus, but was there a reason to teach it to Algebra I students? Well, I don't recall going too far in depth on this, but the only reason to learn completing the square in Algebra I is to derive the quadratic formula. So how do you get the quadratic formula by completing the square? Well, we know the quadratic formula works for all quadratic equations, so we start with our generic equation. But there's an important rule for completing the square. Do you remember it? The number in front of x-squared has to be one (1). The problem here is that a can be any number (except 0). Fortunately, since a cannot be 0, we can divide everything in the problem by a.

This makes the number in front of x-squared one (1). And zero divided by a will simplify to zero (0).

The next step is to subtract the constant term (the one without an x). Here, that happens to be c/a. Remember, since this is an equation, we have to do everything to both sides equally.

We now have the problem below.

This is the trickiest part of completing the square. We now have to add a number to both sides, but not just any number. We want to take the coefficient (the number in front) of x, divide it by 2 and then square it. In symbols, it would look like this:

In this case, the coefficient of x is b/a. So we put b/a in the empty space.

We can rewrite the inside as b/2a. We can then square it.

We now have the term we want to add to our equation.

Since we're dealing with the equation, we add our new term to both sides.

To make everything easier later, we want to combine both terms in the right-hand side. To do so, we must first find the common denominator. Here, it's 4-a-squared. This makes c/a into 4ca/4-a-squared.

We then combine the two terms into one fraction.

Now to deal with the left-hand side. This method is called completing the square, and that's what we're trying to do. Notice the x added to a blank below? That whole term will be squared, and it's the square we're trying to complete.

But what number goes there? Well, remember the number we squared earlier? We want to use the number before it was squared (b/2a). Notice I'm not adding anything to the right-hand side. This is because we are not adding b/2a; we are only rewriting the left-hand side.

The reason we wanted to complete the square is so we could take a square root. Since we're taking the square root (and changing the left-hand side), we have to do it to both sides. This is important: make sure to include ONE plus/minus sign. Since we are introducing a square root to the problem, we have to remember to include the plus/minus sign. It makes the most sense to put it on the right, so that's where it'll go.

We simplify the left-hand side.

On the right-hand side, we can simplify our radical by taking the square root of 4-a-squared to get 2a on bottom.

Now we want to get x alone (since we're solving for x). We have to subtract b/2a from both sides.

This gives us x by itself on the left. Now, we combine our two terms on the right-hand side. This leaves us with the quadratic formula, in all it's messy glory!

I don't remember ever learning this in Algebra I, but I think it would've helped if we did. I like to know why we learn what we do (completing the square) and where they come from (the quadratic formula), and I'm sure know both sides would help some students learn. If a student can understand completing the square AND can work with variables, he or she can follow a lesson on this. This derivation is simple enough that it can be given as enrichment work for students who finish early with class work or a test.

Anyway, I hope you got something out of this. If not, well, there's always next time.

If you have any questions, or if anything isn't clear, just let me know in the comments.

 

I first saw quadratic equations during my eight grade year from my Algebra I and Math Team teacher Tiffany Brooks. 

The images used here were produced by me using Noteshelf for iPad.