 So, remember factoring is the hardest easy problem in mathematics, so we'll take a look at a number of ways to factor, and one of them relies on finding the difference of squares. So to factor a quadratic polynomial, we can use the quadratic formula to find the roots, and then the root theorem to find the factors. But you might object to this. You might say, hey, that's too easy, and that works on every quadratic. Isn't there a harder way to do things that will work on every quadratic? And the answer is, yes, yes there is. So we'll take a look at a harder method that will work on every quadratic. To proceed, we'll find a special product, a minus b times a plus b. So we'll find that product. This gives us an important result. A minus b times a plus b is equal to a squared minus b squared. For example, let's try to factor x squared minus 49. So let's make a few observations. First, the first term, x squared, is a square. It is the square of x. Yeah, I know that's not exactly world shaking. What is important to recognize is that our second term, 49, is also a perfect square. It's 7 squared. So this is a difference of squares. And because it's a difference of squares, we can write down its factors immediately. One way to remember this is that this is the difference of the square roots times the sum of the square roots. So since this is x squared minus 7 squared, this will factor as x minus 7 times x plus 7. Now in a kind and gentle universe, or with a kind and gentle math teacher, you'd always have differences of squares to factor. Well, the universe isn't kind and gentle, and neither am I. But this is math. And if there's only one thing you learn in this course, you'll probably fail it. But among the things you should learn is that you can have anything you want as long as you pay for it. So if you want a square, make a square. We'll do that by completing the square. So isolate the constant term. And let's consider these variable terms. That first term is a square. Our second term is 2 times 2 times x. And so that means to complete the square, we need to add 2 squared, otherwise known as 4. Now it's important to understand this is not an equation. So if I add 4, I have to pay for it later by subtracting 4. It's important to keep this in mind. If we change the left-hand side here, we're no longer trying to factor x squared plus 4x minus 12. We're trying to factor something else. And that may be useful, but it's not what the question is asking us to do. So we'll complete the square by adding 4 and paying for it by subtracting 4. So now our first set of terms is a perfect square, x plus 2 squared. We also have minus 12, minus 4. Well, that's really the same thing as minus 16, and that's also a perfect square, 4 squared. And so now I have a difference of squares, which means I can factor it as the product of the difference and sum of the corresponding square roots. So this becomes x plus 2 minus 4 times x plus 2 plus 4. And we can clean up some of the arithmetic. This plus 2 minus 4 is really the same thing as minus 2, and this plus 2 plus 4 is really the same thing as plus 6, and here's our factorization. So let's try to factor x squared plus 12x minus 24. So first, we'll isolate the constant term, then we'll complete the square. So of our terms, the first term is a square, x squared. We'd like our second term to be 2 times x times something, and if our something is 6, then that will work. And so that says that we need to add 6 squared and pay for it later by subtracting 6 squared, also known as 36. And so my expression becomes x squared plus 12x plus 6 squared minus 24 minus 36. And so my first set of terms is a perfect square, x plus 6 squared. My remaining terms, well, that's really the same as subtracting 60, and I do need to write this as a difference of squares, so my first term is a square. My second term can be rewritten as the square root of 60 squared. And so now I have a difference of squares, so I can factor it as x plus 6 minus square root of 60 times x plus 6 plus square root of 60. While this is a factorization, we do note that x plus 6 minus square root of 60 has non-rational terms. Square root of 60 is not a rational number, and so this is not a rational factor, and this expression can't be factored over the rationals. We should say so. Let's take a look at another example. So again, we'll isolate the constant term and complete the square on the first two terms. So our first term, 4x squared, is a square. It's the square of 2x. Our second term, minus 12x, we want that to be 2 times something times 2x, and so our something has to be negative 3. So we need to add and subtract negative 3 squared, otherwise known as 9. So if I add and subtract 9, my first set of terms is a perfect square, 2x minus 3 squared. And my second set of terms is 0, and I really don't have to carry around a plus 0. And, well, actually we've written this as a product, so we factored our expression. Or take a look at something like this. We'll complete the square. So again, we'll isolate our constant term. Our first term is a square, x squared. Our second term, minus 5x, we want to write this as 2 times something times x, and so we can do that if our something is negative 5 halves. So we need to add and subtract negative 5 halves squared, otherwise known as 25 fourths. Now this does force us to deal with fractions, but the sad fact of life is, you can run, but you can't hide. Fractions are inevitable. You should get comfortable using them. So we'll add and subtract 25 fourths, and that makes our first set of terms a perfect square. Our second set of terms works out to be 121 fourths, which is also a perfect square, which is now a difference of squares. And we can factor x minus 5 halves minus 11 halves times x minus 5 halves plus 11 halves, and that simplifies. The obvious question to ask at this point is, is there an easier way? And the answer is, no. And that shouldn't surprise us. Factoring is the hardest easy problem in mathematics. So there really aren't any easy ways to factor, but there are harder ways to factor, and we'll take a look at those next.