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## Algebra 1

### Course: Algebra 1>Unit 13

Lesson 8: Factoring quadratics with perfect squares

# Factoring perfect squares: shared factors

Sal finds the binomial factor shared by 4x^2+12x+9 and 4x^2-9.

## Want to join the conversation?

• At , why did he just ignore 2*2*3x and not include it in his factoring?
• He hasn't actually ignored the 2*2*3x. Remember, factoring a quadratic expression is basically reversing FOIL (First, Outer, Inner, Last). He factored each term in 4x^2 + 12x+9, noticing that the first and last terms are perfect squares. The first term = 2x times 2x. the last term = 3 times 3. If this expression = the perfect square of one binomial, then the middle term will equal 2 times the first term times the last term. This is the Outer and Inner parts of FOIL. In symbols, 12x = 2(2x times 3). It does.
This tells him that 4x^2 + 12x + 9 = (2x+3)^2
Test the result by applying FOIL to (2x+3)(2x+3). I found that doing this while really thinking about what is happening helped me to understand factoring quadratic equations.
• The energy points just show how many videos you have watched and questions you have answered. It is a way of "gamifying" and encouraging users to keep learning more.
• in he said that it was -3^2. -3^2=-3*-3=9≠-9. -9 was up there, where as 9 wasn't. Why is that, and why did he do that?
• Sal has a negative 9. To have a negative 9, the factors must be one negative and one positive.

Also, (-3)^2 = -3(-3) = +9
When you have -3^2, the exponent applies only to the 3, not -3. Thus: -3^2 = -(3*3) = -9

Hope this helps.
• I am more comfortable with factoring quadratics by grouping. Is that okay? Is there any type of special circumstance that you have to factor using the perfect squares?
• Yes, factoring by grouping would work. However, it can take longer that using / recognizing that you have a perfect square trinomial and using the pattern to do the factoring.

There are also later topics / concept that require that you understand the relationship between perfect square trinomials and their factors. There is a process called "completing the square" that leverages this relationship. It is used to solve quadratic equations and to form / understand the equation of a circle.
• What is the FOIL method?
• This is a way to remember how to multiply two binomials.
Have a look in the Algebra 1 material
under Introduction to Polynomials
in the 4th video in the section Multiplying Binomials.
• Is it important when someone says, "sounds like a little bit of voodoo."?
(1 vote)
• It means it sounds different/unfamiliar (I think)
(1 vote)
• What about (a+b)^3 and (a+b)^4,(a+b+c)^2,(a+b-c)^2,(a-b-c)^2,a^3+b^3,a^4+a^4 and a^3-a^3?
Where are explanations video?
(1 vote)
• Right now, you should only really worry about quadratics and the squares of binomials.

(a+b)^2 = a^2+b^2+2ab.

When the time comes, you'll learn about polynomials to the third degree and whatnot.
(1 vote)
• Can any positive number have negative factors? And x= 16^1/2 gives +4/-4 or both
(1 vote)
• Wait is the answer to the problem 2x + 3 or (2x + 3) or does it matter?
(1 vote)
• It doesn't matter. The parentheses only matter when the binomial is part of a larger expression.
(1 vote)
• Is there a video for factoring perfect squares that have a negative sign in it, like 3x^2 -24x +48? If not would someone be able to show me how to do that? Thank you.
(1 vote)

## 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.