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## Factoring quadratics with perfect squares

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# Factoring perfect squares: shared factors

CCSS Math: HSA.SSE.A.2, HSA.SSE.B.3, HSF.IF.C.8

## Factoring quadratics with perfect squares

## Video transcript

- [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.