Factoring quadratics with perfect squares
- [Narrator] We're going to learn to recognize and factor perfect square polynomials in this video. So for example, say I have the polynomial x squared plus six x plus nine. And then someone asks you, "Hey, can you factor this "into two binomials?" Well, using techniques we learned in other videos, say, "Okay, I need to find two numbers "whose product is nine and whose sum is six." And so I encourage you to pause this video and say, "Well, what two numbers can add up to six, "and if I take their product I get nine?" Well, nine only has so many factors, really one, three, and nine. And one plus nine does not equal six. And so, and negative one plus negative nine does not equal six. But three times three equals nine, and three plus three does equal six. Three times three, three plus three. And so we can factor this as x plus three times x plus three, which is of course the same thing as x plus three squared. And so what was it about this expression that made us recognize, or maybe now we will start to recognize it as being a perfect square? Well, I have of course some variable that is being squared, which we need. I have some perfect square as a constant, and that whatever is being squared there, I have two times that as the coefficient on this first degree term here. Let's see if that is generally true. And I'll switch up the variables just to show that we can. So let's say that I have a squared plus 14 a plus 49. So a few interesting things are happening here. All right, I have my variable squared. I have a perfect square constant term, that is seven squared right over here. And my coefficient on my first degree term here that is two times the thing that's being squared. That is two times seven, or you can say it's seven plus seven. So you can immediately say, "Okay, "if I want to factor this, "this is going to be a plus seven squared." And you can of course verify that by multiplying out, by figuring out what a plus seven squared is. Sometimes when you're first learning this, you're like, "Hey, isn't that just a squared "plus seven squared?" No! Remember, this is the same thing as a plus seven times a plus seven. And you can calculate this by using the foil, F-O-I-L technique. I don't like that so much because you're not thinking mathematically about what's happening. Really you just have to do distributive property twice here. First you can multiply a plus seven times a. So a plus seven times a. And then multiply a plus seven times seven. So plus a plus seven times seven, and so this is going to be a squared plus seven a, plus, now we distribute the seven. Plus seven a plus 49. So now you see where that 14 a came from. It's from the seven a plus the seven a. You see where the a squared came from. And you see where the 49 came from. And you can speak of this in more general terms. If I wanted to just take the expression a plus b and square it, that's just a plus b times a plus b, and we do exactly what we did just here, but here I'm just doing in very general terms with a or b and you can think of a as either a constant number or even a variable. And so this is going to be, if we distribute this, it's going to be a plus b times that a plus a plus b times that b. And so this is going to be a squared, now I'm just doing the distributive property again. A squared plus ab plus ab plus b squared. So it's a squared plus two ab plus b squared. So this is going to be the general form. So if a is the variable, which was x, or a in this case, then it's just going to be whatever squared and the constant term is going to be two times that times the variable. And I want to show that there's some variation that you can entertain here. So if you were to see 25 plus 10x plus x squared, and someone wanted you, said "Hey, why don't you factor that?" we could say, "Look, this right here is a perfect square. "It's five squared. "I have the variable squared right over here, "and then this coefficient on our first degree term "is two times five." And so you might immediately recognize this as five plus x squared. Now of course you could just rewrite this polynomial as x squared plus 10 x plus 25 in which case you might say, "Okay, variable squared, "some number squared, five squared, "two times that number is the coefficient here. "So that's going to be x plus five squared." And that's good because these two things are absolutely equivalent.