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Difference of squares intro

When an expression can be viewed as the difference of two perfect squares, i.e. a²-b², then we can factor it as (a+b)(a-b). For example, x²-25 can be factored as (x+5)(x-5). This method is based on the pattern (a+b)(a-b)=a²-b², which can be verified by expanding the parentheses in (a+b)(a-b).

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  • hopper cool style avatar for user Ritvik‮
    At , why is it called the Foil method? Does 'Foil' stand for something or is it just called that?
    (37 votes)
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    • leaf green style avatar for user kubleeka
      FOIL stands for "First, Outside, Inside, Last". You multiply together the first term in each binomial, the outside (leftmost and rightmost) terms, the inside terms, and the last term in each binomial. Take those four products, add them up, and you have the expanded expression.
      (82 votes)
  • hopper jumping style avatar for user helen
    What if we have a difference of squares like x^4 - y^4?
    (15 votes)
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  • winston baby style avatar for user Davidroe1
    Well hello there friendly neighborhood precal student here
    just wondering if someone can recommend me to a video or provide a little help here
    the problem is x^2-10x-24
    i had moved the 24 over from the right side
    i have no idea what else to do afterward since all of these videos only explain it with an x^2 and a lone number

    please help
    thank you :)
    (12 votes)
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    • duskpin ultimate style avatar for user Pei-Hsin Lin 林培心 🌸
      The way to tell when you don't have a difference of 2 squares is if you can't find two perfect squares that are connected with a subtraction sign (the difference part of the name).

      The following are valid examples of a difference of two squares, which can always be factored into a pair of conjugates (a ± b).
      Notice that each term is a perfect square, and each starting expression is a mathematical difference:
      25x² - 49y² factors into (5x + 7y)(5x - 7y)
      z⁴ - ¼ factors into (z² + ½)(z² - ½)
      9 - p² factors into (3 + p)(3 - p)

      As far as the expression that you were trying to factor x² - 10x - 24 = (x + ?)(x - ?) see below:
      * The two ?s must be factors of -24. One's positive and one's negative so that their product is a negative number, the -24.
      * Furthermore, they must sum to -10, the coefficient of the x term, so that the OUTER product and INNER product sum to the -10x.
      * 2 and -12 satisfy these conditions, so x² - 10x - 24 = (x + 2)(x - 12).
      * Lastly, the factorization can be VERIFIED with binomial multiplication (FOIL - First Outer Inner Last):
      (x + 2)(x - 12) = 
      x∙x - 12∙x + x∙2 - 12∙2 =
      x² - 12x + 2x - 24 =
      x² - 10x - 24
      (24 votes)
  • primosaur ultimate style avatar for user Steve
    One question... When you do have a difference of perfect squares problem...do you HAVE to use the a^2-b^2 method? Or can you still use the GCF method for them? It seems like every example I see, always uses the A^2-B^2 way to factor the difference of perfect squares...but the GCF way would work as well right?
    (5 votes)
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  • piceratops ultimate style avatar for user JasonMasc
    Is Difference of Squares is helpful when you try to find equations of conic sections?
    (6 votes)
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  • starky seed style avatar for user Strantz Kendall
    when people write this symbol ^ what is it? Is it multiplication or divions
    (1 vote)
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  • leafers tree style avatar for user Yousif
    0:5 till maybe. Does a difference of squares have to be subtraction or can it be addition, because judging of the name "difference". So I'm thinking subtraction only, but I just want it clarify whether it can also be addition
    (3 votes)
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    • mr pink green style avatar for user David Severin
      Difference means subtraction, so it can only be a subtraction. If it is an addition such as x^2 + 1, think what that would do to the discriminant, b^2 - 4ac = 0^2 - 4(1)(1) = -4 which would mean you would have to take the root of a negative number, not allowed in the real domain. With c being a negative perfect square, the determinant is 0^2 - 4(a)(-c) so two negatives cancel, and since each of 4 and a and c are perfect squares, the root would be a whole number.
      (2 votes)
  • aqualine ultimate style avatar for user J'immy J'onzz
    Does the difference of squares method still work if it was (U^2)+(V^2) instead of (U^2)-(V^2)? If it doesn't work then why not?
    (2 votes)
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    • starky ultimate style avatar for user KLaudano
      The difference of squares factoring only works for (u^2)-(v^2), that is why it is called the difference of squares. (There is a way to factor (u^2)+(v^2), but it requires complex numbers. (u^2)+(v^2)=(u+v*i)*(u-v*i))
      (3 votes)
  • piceratops ultimate style avatar for user Danny Li
    Why did you use the distributive property twice?
    (2 votes)
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  • aqualine seed style avatar for user Thomas B
    How do you figure out the width or length of a rectangle, this is very confusing for me as there was nothing in the videos about it. Please help me. Thanks
    (2 votes)
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Video transcript

- [Instructor] We're now going to explore factoring a type of expression called a difference of squares and the reason why it's called a difference of squares is 'cause it's expressions like x squared minus nine. This is a difference. We're subtracting between two quantities that are each squares. This is literally x squared. Let me do that in a different color. This is x squared minus three squared. It's the difference between two quantities that have been squared and it turns out that this is pretty straightforward to factor. And to see how it can be factored, let me pause there for a second and get a little bit of review of multiplying binomials. So put this on the back burner a little bit. Before I give you the answer of how you factor this, let's do a little bit of an exercise. Let's multiply x plus a times x minus a where a is some number. Now, we can use that, do that using either the FOIL method but I like just thinking of this as a distributive property twice. We could take x plus a and distribute it onto the x and onto the a. So when we multiply it by x, we would get x times x is x squared, a times x is plus ax and then when we multiply it by the negative a, well, it'll become negative a times x minus a squared. So these middle two terms cancel out and you are left with x squared minus a squared. You're left with a difference of squares. x squared minus a squared. So we have an interesting result right over here that x squared minus a squared is equal to, is equal to x plus a, x plus a times x minus a. And so we can use, and this is for any a. So we could use this pattern now to factor this. Here, what is our a? Our a is three. This is x squared minus three squared or we could say minus our a squared if we say three is a and so to factor it, this is just going to be equal to x plus our a which is three times x minus our a which is three. So x plus three times x minus three. Now, let's do some examples to really reinforce this idea of factoring differences of squares. So let's say we want to factor, let me say y squared minus 25 and it has to be a difference of squares. It doesn't work with a sum of squares. Well, in this case, this is going to be y and you have to confirm, okay, yeah, 25 is five squared and y squared is well, y squared. So this gonna be y plus something times y minus something and what is that something? Well, this right here is five squared so it's y plus five times y minus five and the variable doesn't have to come first. We could write 121 minus, I'll introduce a new variable, minus b squared. Well, this is a difference of squares because 121 is 11 squared. So this is going to be 11 plus something times 11 minus something and in this case, that something is going to be the thing that was squared. So 11 plus b times 11 minus b. So in general, if you see a difference of squares, one square being subtracted from another square and it could be a numeric perfect square or it could be a variable that has been squared that can be, that you could take the square root of. Well, then you could say, alright, well, that's just gonna be the first thing that squared plus the second thing that has been squared times the first thing that was squared minus the second thing that was squared. Now, some common mistakes that I've seen people do including my son when they first learned this is they say, okay, it's easy to recognize the difference of squares but then they say, oh, is this y squared plus 25 times y squared minus 25? No, the important thing to realize is is that what is getting squared? Over here, y is the thing getting squared and over here it is five that is getting squared. Those are the things that are getting squared in this difference of squares and so it's gonna be y plus five times y minus five. I encourage you to just try this out. We have a whole practice section on Khan Academy where you can do many many more of these to become familiar.