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Multiplying binomials: area model

Sal expresses the area of a rectangle whose height is x+2 and width is x+3.

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  • aqualine ultimate style avatar for user jaysonm24476
    why do I have to do math.
    (53 votes)
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    • aqualine ultimate style avatar for user Hannah Alisse
      That's an interesting question. I had to use google to get some adequate answers:

      Math is good for your brain! It makes you smarter (insert sarcasm: wow really? I didn't know that), and improves your problem solving abilities.

      When you become an adult, you'll have to manage money. Doing that requires a lot of math. Even just going to the grocery store and shopping smart takes basic arithmetic (and basic algebra if you want to be meticulously thorough).

      Math helps you understand the world! This is why I really like math (note: I like math as a concept. Calculations are boring :P) Everything around you. Every tree, flower, bookcase, speck of dust, it has math woven into it's being. As Galileo said, "The universe is a book, and math is the language it's written in." Now admittedly, the math you're learning right now is kind of boring, but if you stick with it, you'll be able to learn some really cool concepts!

      If you don't want to stick with it, and just want some simple math awesomeness, there's a youtuber I really like, ViHart, that explains some really fun mathematical concepts without using numbers! If you're interested, I highly recommend checking it out.

      Link to her videos:
      https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart

      Anyway, that answer was really cheesy, and you've probably heard it a million times already, but yeah. Math is fun, but it can get REALLY annoying sometimes, especially when you're trying to learn this thing, and the thing makes no sense, and the thing is stupid an you're like "I DON'T WANT TO LEARN THIS THING." I've been there. Many times. But you'll get through it :)
      (74 votes)
  • duskpin tree style avatar for user calculator
    so frustrating with 9th grade math (╯▔皿▔)╯
    (18 votes)
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  • duskpin sapling style avatar for user Claria
    At why can you combine 3x and 2x into 5x, but you can't combine x-squared with those?
    (4 votes)
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  • male robot hal style avatar for user RN
    You can just factor out the binomials using the FOIL method right?
    (6 votes)
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  • blobby green style avatar for user abner3
    is this important
    (6 votes)
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  • blobby green style avatar for user matanwx135
    but... but... this is a parabola! how come the area of this squares combined can be equle to n ENDLESS line that only intersects the x line 2 times? this is INSANITY!
    (3 votes)
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    • leaf green style avatar for user Katriana
      I think you are saying that the equation he ends up with could also be the equation defining a parabola, correct? Well, that might not be quite so strange as you think. You see, x is a variable, so we do not know its value. Depending on the value of x, the area of the square will be different. You could make a graph using different x values in this equation, and the y values you come up with will be the area of the square corresponding with different x values. What's more, if you plot the points (the x values and the area values that go with them), you will create a parabola!

      I am so glad you mentioned this because I would not have otherwise thought about this interesting relationship. Math is so amazing, isn't it?

      (The negative values wouldn't really make sense for length and area, though.)
      (10 votes)
  • blobby green style avatar for user Ethan
    how is this gonna help me in life and prepare me for the future
    (3 votes)
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  • old spice man green style avatar for user Miguel McKinnon
    Hey Khan Academy, I'm a bit confused with a multiplying binomials question I got wrong on a quiz.

    I was under the impression, based on Sal's multiplying binomials videos, that if you were to multiply a binomial such as: (a+7y)(9y-12a) that you can distribute the second binomial to the first binomial: 9y(a+7y)-12a(a+7y) and solve.

    That's how I've been approaching the multiplication of binomials and I've been getting the correct answers.

    The question on the quiz distributred the first binomial to the second, so the reverse order to what I thought Sal was instructing us to do.

    Any clarity you can provide would be much appreciated!

    Thanks,
    (5 votes)
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    • leaf orange style avatar for user A/V
      Both ways are correct, whether you start from the first binomial or the second binomial, you still end up with the same answer.

      To prove:
      (x+2)(x-3)
      Can be written as:
      x(x-3)+2(x-3)
      x²-3x+2x-6
      x²-x-6
      Can also be written as:
      x(x+2)-3(x+2)
      x²+2x-3x-6
      x²-x-6
      (3 votes)
  • blobby green style avatar for user AlekH
    Wow I just love math
    (4 votes)
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  • mr pants teal style avatar for user Qasim Hashmi
    i didn't understand why you can't combine X^2 with 3x
    (2 votes)
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Video transcript

- [Voiceover] So I have this big rectangle here that's divided into four smaller rectangles, and what I wanna do is I wanna express the area of this larger rectangle and I wanna do it two ways. The first way I wanna express it as the product of two binomials and then I wanna express it as a trinomial, so let's think about this a little bit. So one way to say well look, the height of this larger rectangle from here to here, we see that that distance is x, and then from here to here it's two, so the entire height right over here, the entire height right over here is going to be x plus two. So the height is x plus two, and what's the width? Well the width is, we go from there to there is x, and then from there to there is three, so the entire width is x plus three. x plus three. So just like that, I've expressed the area of the entire rectangle, and it's the product of two binomials. But now let's express it as a trinomial. Well to do that, we can break down the larger area into the areas of each of these smaller rectangles. So what's the area of this purple rectangle right over here? Well the purple rectangle, its height is x and its width is x, so its area is x squared. Let me write that, that's x squared. What's the area of this yellow rectangle? Well it's height is x, same height as right over here, its height is x and its width is three, so it's gonna be x times three, or 3x. It'll have an area of 3x. So that area is 3x, if we're summing up the area of the entire thing, this would be plus 3x. So this expression right over here, that's the area of this purple region, plus the area of this yellow region, and then we can move on to this green region. What's the area going to be here? Well the height is two and the width is x, so multiply height times width is gonna be two times x, and we can just add that, plus two times x, and then finally this little grey box here. Its height is two, we see that right over there, its height is two, and its width is three, we see it right over there, so it has an area of six. Two times three. So plus six, and you might say well this isn't a trinomial, this has four terms right over here, but you might notice that we can add, that we can add these middle two terms, 3x plus 2x, if I have three x's and I add two x's to that, I'm gonna have five x's. So this entire thing simplifies to x squared, x squared plus 5x plus six. Plus six. So this and this are two ways of expressing the area, so they're going to be equal, and that makes sense 'cause if you multiply it out, these binomials, and simplify it, you would get this trinomial, and we can do that really fast. You multiply the x times the x, actually let me do that in the same colors. You multiply the x times the x, you get the x squared. You multiply this x times the three, you get your 3x. You multiply the two times the x, you get your 2x. And then you multiply the two times the three and you get your six. So what this, I guess you can say this area model does for us is it hopefully makes a visual representation of why it makes sense to multiply binomials the way we do, and in other videos we talk about it as applying the distribution property twice, but this gives you a more visual representation for why it actually makes sense.