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Factoring difference of squares: analyzing factorization

Sal analyzes two different factorizations of 16x^2-64 and determines whether they are correct.

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  • piceratops ultimate style avatar for user Andrew Myrold
    Which expression is further simplified?
    (9 votes)
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    • hopper happy style avatar for user Sheng
      To sum it up, (4x-8)(4x+8) is not the most simplified because you can still factor out a four --> (4x is divisible by 4 and 8 is also divisible by 4). 16 (x - 2)(x + 2) is the most simplified because 2 can be divided by 2 but x (which is the same as 1x) can't be divided by any number other than one.
      (21 votes)
  • female robot grace style avatar for user BADA DUM BADA DUM
    How will we ever use this in real life?
    How is this information ever going to help us?
    What's the purpose of this particular topic being here on this website for us to study so we can use it in our tests and exams?









    I'm not trying to criticize anyone, i'm just curious.
    (5 votes)
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  • winston baby style avatar for user Michelle Young
    Hi, since there are squares, how can -1 be squared. Isn;t that an imaginary number? How can you find the square root of an imaginary number? Please help!
    (3 votes)
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  • aqualine ultimate style avatar for user Charles  Paredes
    Why at Sal says it doesn't make any sense?
    (2 votes)
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  • starky ultimate style avatar for user aryan0904
    For these types of problems can't you just foil the numbers?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      Yes, you can use FOIL to multiply the 2 binomials.
      However, there are certain binomials that create what are called "special products". It is important to learn and understand them as it makes other tasks in Algebra a lot easier to do. Sal is trying to teach you one of these special products in this video. Specifically, he is showing you the pattern that creates a difference of 2 squares.
      (3 votes)
  • old spice man green style avatar for user David
    I am confused. I tried to calculate what Moussa would get using the equation 16 (x+2) (x-2). The order of operations tells me that the first thing I should do is multiply 16 by the terms in the first set of parentheses: x + 2. Doing so brings the equation to 16x + 32 (x - 2). The order of operations tells me that the next thing I should do is multiply 32 by the terms in the remaining set of parentheses: x - 2. Doing so brings the equation to 16x + 32x - 64. I can add the first two terms in the equation since they both feature the same variable. Doing so gets me 48x - 64. I don't understand why Sal didn't follow the order of operations like I did.
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      You started out correctly.... but you lost a set of parentheses.
      16 (x+2) (x-2) = (16x + 32) (x-2)
      Multiply the 16 with the binomial x+2 doesn't change the fact that the entire binomial needs to get multiplied with the 2nd binomial. So, you can't drop the parentheses.
      Hope this helps.
      (5 votes)
  • blobby green style avatar for user Shymirah Morrison
    I didnt understand the video...i could barely hear or keep up with anything going on in the video
    (2 votes)
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  • mr pants purple style avatar for user spoonerP1
    is 4(2x+4)(2x-4) also a valid answer?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      Yes, your factors work. You can verify this yourself by multiplying them and verify that you get back to the original polynomial.

      But - If you were asked to completely factor the polynomial, your version would be incomplete. Fatu's in the video is also incomplete. The reason is that both your version and Fatu's contain common factors that should be factored out. Moussa's version would be correct for the polynomial to be completely factored.
      (1 vote)
  • aqualine ultimate style avatar for user averymills7
    How do you do 16-49(x-squared)
    (2 votes)
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  • blobby green style avatar for user 1693891
    Can we have differences of square for 1 like we can have for -1
    (1 vote)
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    • mr pink green style avatar for user David Severin
      Difference by definition means subtraction, so adding x + 1 is not a difference, and there is no such thing as the sum of squares. Think of what that would do to the determinant of the quadratic formula, b^2 - 4ac. Since b would not exist, -4ac would be a negative number as long as a and c are both positive integers, so there would be no solutions.
      (1 vote)

Video transcript

- [Voiceover] Moussa and Fatu were each asked to factor the quadratic expression 16 x-squared minus 64. Their responses are shown below. So Moussa factored it this way. Fatu factored it this way. Which student wrote an expression that is equivalent to 16 x-squared minus 64? So I encourage you to pause the video and figure that out. Which student wrote an expression that is equivalent to our original one, 16 x-squared minus 64? Well let's work through it together, so let's see if first we can factor this out somehow to get what Moussa got and it looks like Moussa first factored out a 16 and then he was left with a difference of squares. So let's see if we can do that. So, we can write our original expression. 16 x-squared minus 64, we can write that as 16 times x-squared minus 16 times four. And when you write it like that, it's very clear that you can factor out a 16. So this is going to be equal to 16 times what you have left over is x-squared minus four and then x-squared minus four, that's a difference of squares right over there. So, that part we can factor as, so we have our original 16 and then... this part right over here, we can write as x plus two times x minus two. x plus two times x minus two. If what I just did in this last step, going from x-squared minus four to x plus two times x minus two doesn't make any sense, I encourage you to watch some of the introductory videos on factoring and difference of squares. But the basic idea, I have a form here of a-squared minus b-squared, so it's going to have the form of a plus b times a minus b and in this case it's x-squared minus two squared. So it's going to be x plus two times x minus two. So that's exactly what Moussa got. So this one, so Moussa, did get an expression that is equivalent to 16 x-squared minus 64. Now let's think about Fatu. So Fatu didn't factor out a 16 from the get-go. It looks like he just immediately recognized that our original expression is itself a difference of squares even if we don't factor out a 16, and so let's re-write it. So our original expression, we could write as, so instead of writing, well I'm just going to write it like this, this is our original expression. 16 x-squared minus 64. That's the same thing as, 16 x-squared is the same thing as four x, the whole thing squared and then minus eight squared. So when you write it like this, it's clear that this is a difference of squares, so this is going to be four x plus eight times four x minus eight. Four x plus eight times four x minus eight. Once again, if this last step that I did doesn't make a lot of sense I encourage you to watch the video on factoring difference of squares where we go a lot more into the intuition of it. But when you see it this way you realize that Fatu also got an expression that is equivalent to 16 x-squared minus 64, so they both did.