 So let's talk about completing the square algebraically. So this is one of the more useful algebraic skills and the basic idea is this, suppose I have the x squared and x terms of a quadratic polynomial. What I'd like to do is I'd like to add some constant that's going to make it a perfect square trinomial, a perfect square. So it's going to be helpful to remember what those perfect squares look like. So if I have x plus h squared, I'll expand it and I get x squared plus 2hx plus h squared. Likewise, if I have x minus h squared, I'll expand it and I'll get x squared minus 2hx plus h squared again. And that means if I have an expression like x squared plus 8x, then comparing this, well, x squared plus 8x looks like the first two terms of this, which means that this thing must come from the square of something like x plus h. And likewise, if I have a difference x squared minus 5x, well, that looks like these two terms here. And so this must come from the square of something like x minus h squared. Now, let's see how that works out. In both cases, we have to figure out what our value of h is. So for example, let's say I have x squared plus 8x and I want to find a constant that I need to complete the square. So since it's the sum, it's got to come from the square of x plus h and I have to figure out the value of h. So paper is cheap. Let's set down our two expressions and compare them. So x plus h squared, I'm going to compare that to x squared plus 8x. So I'll expand out x plus h squared, x squared plus 8x, I don't need to do anything with. Here's my expansion, x squared plus 2hx plus h squared. And I'll compare my perfect square with what I have. So both of them have an x squared. That's good. The perfect square has 2hx and the expression that we have has an 8x. So if I compare those two, 2hx to 8x, that says that 2h and 8 must be the same thing. So that tells me h equals 4. Well, let's fill that in. If I know that h equals 4, I can fill that in. And I get x squared plus 2 times 4x plus 4 squared. And I'll let some of the dust settle here. That's x squared plus 8x plus 16. And again, I'll compare my two expressions. They have x squared, they have 8x. My perfect square has a 16, and my original expression does not. So the constant that I need to complete the square is going to be 16. That's the thing that I'm missing. Well, let's take a look at another example, x squared minus 5x. So here I have a difference, and so this difference must have come from the square of x minus h for some h to be determined. And so I'll, again, set down my two expressions and compare them. x minus h squared versus x squared minus 5x. And again, I'll expand out that x minus h, and I'll compare my two expressions. So again, I have 2hx here, I have 5x here, and I'll compare my two expressions. That says 5 and 2h must be the same thing. And so that tells me h equals 5 halves, and I'll substitute that in. So x squared minus 2h, x plus h squared, I'll do a little bit of algebra. And what I have, again, comparing my two expressions, I have x squared in both, I have minus 5x in both, I have a 25 over 4 in the perfect square, and I'm missing that in my original expression. So what do I need? The constant that I need to complete the square is this 25 over 4.