College Algebra: Proving the Quadratic Formula

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So let's actually look at a very general quadratic equation and use the completing the square method to actually come up with a formula that will always solve quadratic equations and we'll call it the quadratic formula. So I'm going to write down a generic quadratic equation. It would look like this. We'd have an a out in front, ax^2 + bx + c = 0. What I want you to understand is that the variable is x. a, b and c are some numbers but in practice, they may be 4 and q and minus 7 or something. But right now, I'm going to put in just letters to represent those values so we can see a formula that will always work no matter a, b, or c we plug in. So please remember, this is the variable and I want to solve for that and these are just some unknown constants that we'll plug in later.
So now how do you complete the square? The first thing you do is get rid of that coefficient. So if I divide both sides by the a, it cancels here and I'm left with x^2. Now these things are going to get a little messy by the way, times x plus equals 0 because 0 divided a is 0. Now it's going to get a little bit ugly looking but don't let that bother you. We only do this once in our lives so it's worth doing it once.
What's the next step? Take the constant of that and it might not look like a constant to you, but remember, these are supposed to be numbers. It's the x's that we don't know. Take that constant, bring it to the other side. x^2 + times x, a big space, equals; and then I have a . What's the next step in completing the square? What I do is take a look at the coefficient in front of the x, which in this case is a , and I take half of it. So I take half of . That equals and then I square that. So if I square that, I'll see b^2 divided by and if I square the bottom I would see 4a^2. I squared the 2 and I squared the a. That's the value that I'm going to want to add to both sides of this to complete the square. So I'm going to now add to both sides. Adding it to both sides means I'm not changing the equalness of this. Everything balances out perfectly. Just completing the square, I took half of this and squared it. The only difference is that now instead of having like a 7 in here, I have a and so it looks a little bit more threatening. But don't let it threaten you. All we're doing is the exact same idea. This should factor perfectly into a perfect square, squared. We can check that by multiplying this by itself. If you foiled, you'll see x times x, there's that. The inside term is and another . So that's going to give us and this times the last term is going to be . So, in fact, this really is a perfect square.
On this side, what does this give me? Well, I want to get a common denominator here and combined. So I have an a here, but I need a 4a^2. So I'll multiply top and bottom here by 4a. So here I'll see a minus 4ac on top, and the bottom I'll have a 4a^2. Don't forget that term. Plus . So I can actually now combine all these things together. Let me write everything out here. I would have , all squared, equals, and if I combine this, what would it look like? It would look like, well, I've got---just combine the top. I'll write the b first. So that's b^2 minus 4ac. So I would see b^2 minus 4ac, all divided by the common factor of 4a^2. So that's where I am. And now what do I do? I take plus or minus square roots of both sides. That's how you complete the square.
So if I do that, let's see if you can see that a little teeny bit. What I would see is the following. I'd see equals and I'm going to take the square root of the top, plus or minus, the square of the top, b^2 minus 4ac and the square root of the bottom. I only have to take the plus or minus of one of them so I'm going to put the plus and minus in front of everybody. And that's where I am. And now what do it do?
Well, now what I do is I realize that the bottom actually can be written out. I can take the square root of that. The square root of 4 is 2. The square root of a^2 is just a. So, in fact, what I see here is equals plus or minus the square root of b^2 minus 4ac and, all over, just 2a. That's the square root of this. I want x all by itself, so I'm going to bring this term over by subtracting and when I subtract that term, what do I see? Well, if I subtract that term--let me lift this up so you can still see that last equation. That's all that matters right now. What I would see is x equals, well, I'm going to bring this over. It becomes a negative. So plus or minus the square root b^2 minus 4ac all over 2a. Well, look. A 2a on the bottom is the common thing. I can now just combine the tops. So if I combine the tops, what do I see? I see x equals -b plus or minus the square root of b^2 minus 4ac. That's the top and the bottom I see 2a. And so what is the big finish? The big finish is that when I see something like , if we complete the square, we see that x = -b plus or minus the square root of b^2 minus 4ac all divided by 2a. And this will always work no matter what a, b or c is. We could just plug them in and we'll get the 2 solutions. x equals this thing with the plus, x equals this thing with the minus there. And, in fact, this is called the quadratic formula. The quadratic formula because it allows us to solve any quadratic equation. And so, even though it may sort of look intimidating and so forth, you can see the genesis of it. You could see where it came from. It just came from taking a general form of this and just completing the square. That's all.
Next stop we'll take a look at a lot of examples using the quadratic formula. You're not really responsible for being able to derive it like I just did, but at least it gives you a sense of where it came from and I hope that you will be able to memorize this formula because it is extremely powerful and extremely useful.
Equations and Inequalities
Quadratic Equations and the Quadratic Formula
Proving the Quadratic Formula Page [2 of 2]