 So let's take a look at another example of completing the square. Now, fraction-free. Well, almost. Let's contrast this to our traditional method. So say we want to solve 3x squared minus 7x equals 25. We'll solve this in two ways, and first we'll solve it using fractions. And so we can mindlessly remember the formula to complete the square of x squared plus bx add b over 2 squared. And so our first step is getting this expression on the left-hand side into the form of x squared plus something. And so we have to divide everything by 3. And then, do complete the square of x squared plus bx add b over 2 squared. Now, b is our coefficient of x, which is minus 7 thirds. And so, do complete the square of x squared plus bx add b over 2 squared. And so we need to find b over 2 squared, which will be. And so we'll add 4936 to both sides of our equation to get. My right-hand side is the sum of the fractions 25 thirds plus 4936, which we should add. Now, all of this was done so the left-hand side would be a perfect square. So we need to factor the left-hand side. And now we can take the square root of both sides. Don't forget the plus or minus and solve for x. And this is great, provided you're willing to accept two things. First of all, dealing with fractions. And second, blindly following rules without understanding where they came from. And in fact, the role of education in a free society is to make sure the society remains free by constantly asking the question, could we find a better way of doing things? A free society will remain free only as long as people keep asking, there is no better way, do not question the actions of the administration. So while we have this method of completing the square, do complete the square of x squared plus bx add b over 2 squared. It often requires us to deal with fractions. But what if you don't like fractions? And the thing to remember is this, if the only tool you have is a hammer, you must treat every problem like a nail, no matter how bad an idea that might be. And the way forward is this. First of all, remember, don't follow examples, don't memorize procedures, but understand concepts. And the whole idea here is that we'd like to have an expression that is a perfect square. And so if we expand a plus b squared, we get a squared plus 2ab plus b squared. What we want is something that looks like the right-hand side. So let's think about this. We want our first term to be a perfect square. And the first thing to remember is that a squared x squared is a perfect square, so we'll want the coefficient to be a perfect square. And if we multiply the coefficient by itself, it's a perfect square. So let's start out by multiplying everything by 3, the current coefficient of x squared. And so now our first term is a perfect square. Now it helps to look ahead a little bit. We'll want the middle term to be 2ab. So looking ahead, we'll know we'll have to solve 2ab equals minus 21x, but that's going to produce a fraction because we're going to have to divide 21 by 2. But let's think about that a little bit more. If a squared x squared is a perfect square, that also means p squared q squared is a perfect square. If I multiply a perfect square by another perfect square, it's still a perfect square. And what this means is if we multiply everything by 4, we can make everything even and still have a perfect square. So let's also multiply it by 4. And now let's compare. We want our terms to be the start of a perfect square. So we needed our first term to be a squared. So a squared is 36x squared, which means that a is 6x. We'll also want our middle term minus 84x. We want that to be 2ab. And so 2ab is minus 84x. We already know what a is, so we'll replace. And now I want to know what b is. So we'll multiply the 2 and the 6 together to get 12xb. And I'll rearrange the factors a little bit. So now we know that something times x is the same as minus 84 times x. So our somethings have to be the same. 12b has to be the same as minus 84. So we can solve for b. And what I actually need is b squared 49. So we'll add 49 to both sides. And remember, the whole point of this exercise was to get the left-hand side into the form of a perfect square, namely the square of a plus b. So you could try to factor this, but if only there was some way we had to figure out what a and b were directly. Oh wait, we've already found a and b. So we know the left-hand side has to factor as 6x minus 7 squared. And we can take the square root of both sides. Don't forget the plus or minus. And solve for x. So it's really your choice. You can solve this without using fractions. Or you can do complete the square of x squared plus bx and b over 2 squared. Deal with the fractions and end up with a mathematically equivalent answer.