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Factoring quadratics: Perfect squares
- [Voiceover] The quadratic expressions 4x squared, plus 12x, plus 9, and 4x squared minus 9, share a common binomial factor. What binomial factor do they share? And I encourage you to pause the video, see if you can figure it out. So, let's do this by taking each of these expressions, and trying to factor them into binomials, and then see if they share a common binomial factor. I guess they do share one, it's to figure out which one they actually share. So, let's focus first on this 4x squared, plus 12x, plus 9. So, the first thing that might jump out at you is, well let's see, I have a 4 here, this coefficient on the x squared term, that's a perfect square. I could write the entire 4x squared term, I could rewrite that as 2x to the 2nd power, then out here, I have a constant term, the 9, that also is a perfect square. I could rewrite that as 3 squared. And you could say, "Well, gee, does this fit the pattern of a perfect square polynomial?" In order for it to fit the pattern of a perfect square polynomial, the coefficient, here on the x term, would need to be 2 times the product of this 2 and this 3, and it is, indeed, 2 times the product of 2 and 3. It is 2 times 6, so we could write this part, right over here, as 2 times 2, times that 2, times 3, times that 3x, x. Then of course, we have to add these three things together, so plus, plus. And so just like that, we can recognize, hey, this is a perfect square polynomial, right over here. And if what I'm saying, right now, sounds like a little bit of voodoo, I encourage you to watch some of the videos on perfect square trinomials, perfect square polynomials, some of the last two videos in this progression. So this this thing can be rewritten as the same thing as 2x plus 3, 2x plus 3, squared, 2x plus 3, squared. Once again, because it's of the form, you have the entire 2x squared here, you have the 3 squared here, and then this middle term is 2 times the product of these two terms right over here, and so it definitely fit the pattern. So, there you have it. We factored the first expression, and now let's try to factor the second expression. And, immediately when you see this one, it looks like it's a difference of squares, so this one right over there, looks like a difference of squares to me. This, we can rewrite as 2x squared, minus, minus, look at that nice color, minus 3 squared, so minus 3 squared. This is a difference of squares, we've seen multiple times, how to factor difference of squares. If this, again, looks foreign to you, I encourage you to watch those videos, and we explain how that works, and why it works. What is is going to be, when you have something in the form A squared, minus B squared, it's going to be equal to A plus B, times A minus B. So, this is going to be equal to, let me just put the two binomials right over here, so this is going to be A plus B, times A minus B. So, this is going to be 2x plus 3, times 2x minus 3. So, 2x plus 3, times 2x minus 3. And so, what is their common binomial factor? Well, they both involve, when you factor them out, they both have a binomial factor of 2x plus 3. This one right over here, we could rewrite if we want. We could rewrite it as 2x plus 3, times 2x plus 3. That might've been somewhat obvious to you already. So, 2x, 2x, then you have plus 3, plus 3, these two are equivalent. And so you see, we see, that we share in both of these, we share at least one, or we share exactly one, 2x plus 3, so that's the binomial factor that they share: 2x plus 3.