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Squaring binomials of the form (ax+b)²

Sal expands the perfect square (7x+10)² as 49x^2+140x+100. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • winston default style avatar for user Vincent V,
    I don't get it. How is (7x)^2 + (10)^2 wrong?
    (26 votes)
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    • male robot johnny style avatar for user Thomas B
      You can see how it is wrong if you think about it with real numbers instead of x.
      For example, let x = 1.
      Now we have (7+10)^2 which is 17^2=289
      It is NOT 7^2 + 10^2 = 49 + 100 = 149
      If you do it that way you lose the 2 middle terms, in this case 2(7*10), and as you can see, our answer is off by the amount of those terms, 2*7*10 = 140.
      (66 votes)
  • duskpin ultimate style avatar for user Florence Lee
    why is 7x squared+ 10 squared wrong? I don't get it.
    (4 votes)
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    • male robot johnny style avatar for user Thomas B
      You can see how it is wrong if you think about it with real numbers instead of x.
      For example, let x = 1.
      Now we have (7+10)^2 which is 17^2=289
      It is NOT 7^2 + 10^2 = 49 + 100 = 149
      If you do it that way you lose the 2 middle terms, in this case 2(7*10), and as you can see, our answer is off by the amount of those terms, 2*7*10 = 140.
      (21 votes)
  • aqualine ultimate style avatar for user Ingimar Baddi Eydal
    At isn't pretty redundant to point this out this late in the course wouldn't it help to emphasize this point earlier?
    (7 votes)
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  • aqualine tree style avatar for user Darius Totah
    Does this apply to numbers as well, lets say (2+8)^2, or is it only when there are variables involved?
    (4 votes)
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    • stelly orange style avatar for user Lea S.
      It applies to non-variable expressions as well, but it's pretty pointless to use this method on those, since it's needlessly complicated and slow compared to simplifying the expression the traditional way (using the order of operations). See below:

      Using the order of operations:
      (2 + 8)^2 = (10)^2 = 100

      Using this "shortcut":
      (a + b)^2 = a^2 + 2ab + b^2
      (2 + 8)^2 = 2^2 + 2(2)(8) + 8^2 = 4 + 32 + 64 = 100

      So that's why this trick is only a shortcut if variables are involved.
      (4 votes)
  • blobby green style avatar for user Daniel Valdez
    Video

    The (a+b)^2 doesn't match up with the the working example of (7x+10)^2? Yes, that's a question. I'm taking calculus online, which is a nightmare and I am probably the worst at math that you will ever meet.

    Anyways, my dilemma is I'm trying to understand where the 2(7x)(10) is coming from. In the (a+b)^2 example I can follow as the ab+ab is from the distribution. However, the 2(7x)(10) from what I can tell and the lack of clarity besides "you multiply these by 2" doesn't explain why you do that.

    My biggest issue with math is my need to understand something. I excel at physiology and function because I can understand the "why" in something. I am awful at the just do FOIL or use the formula... I'm trying to get better and better at math but I need to know why something is done.
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      Let's start with (a+b)^2. This creates what is called a perfect square trinomial. It is called a special product because there is a specific pattern that squaring a binomial creates. You have 2 choices for simplifying it. You can multiply (FOIL) the 2 binomials (a+b)(a+b), or you can use the pattern.

      When you FOIL: (a+b)(a+b) = a(a) + a(b) + a(b) + b(b) = a^2 + ab + ab + b^2. Notice, the two middle terms are exactly the same. This is always true when a binomial is squared. When you add those 2 terms, you add their coefficients and they create 2ab. Hopefully that helps you see where the 2ab comes from. So, the patter is: (a+b)(a+b) = a^2 + 2ab + b^2.

      Now, let's apply the pattern to (7x+10)^2, Sometimes it helps if you identify what is "a" and "b". In this case: a = 7x and b = 10. This helps you to apply the pattern, because you know what to put in for the variables "a" and "b". Here goes...
      a^2 = (7x)^2
      2ab = 2(7x)(10)
      b^2 = 10^2.
      Put the pieces together and simplify to get the result: (7x)^2 + 2(7x)(10) + 10^2 = 49x^2 + 140x + 100.

      I'm going to use FOIL on the same problem to try to point out how the pattern relates to it.
      (7x+10)(7x+10) = (7x)(7x) + 7x(10) + 10(7x) + 10(10) = (7x)^2 + 70x + 70x + 10^2
      Again, notice the 2 middle terms match. 70x + 70x = 2(70x) = 140x (same as in the pattern).
      Finishing... (7x+10)(7x+10) = 49x^2 +140x + 100.

      Hope this helps.
      (7 votes)
  • leaf blue style avatar for user BRITTANY:)
    would this work for (-5wx^5)^3? how would I do it? thanks
    (3 votes)
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    • old spice man green style avatar for user jmascaro
      Hi Brittany,
      When we have an exponent outside of parenthesis and we are only multiplying or dividing inside the parenthesis, the exponent gets applied to each part of the term. So this gives us:
      -5^3 = -125
      w^3 = w^3
      (x^5)^3 = x^15
      Put it all together and we get
      -125(w^3)(x^15)
      I used parenthesis so that it's easier to read.
      Binomials are different because now we have two terms that we are either adding or subtracting. In this case, we have to use FOIL or some similar method. Example:
      (2x + y^2)^2
      In this case, we have the equivalent of
      (2x + y^2)(2x + y^2)
      So using FOIL we get
      4x^2 + 2xy^2 + 2xy^2 + y^4
      Clean it up by combining like terms and we get
      4x^2 + 4xy^2 + y^4
      Hope this helps your understanding some :-)
      (4 votes)
  • blobby green style avatar for user Jubiemyr Silvio
    at ...(7x)^2 + 2(7x)(10)+ 10^2..I didn't get it...where does the 2(7x)(10) from?I mean can you explain all of it?hehe thank you!
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      When you square a binomial, there are 2 ways to do it.
      1) You use FOIL or extended distribution. 2) You use the pattern that always occurs when you square a binomial. Sal shows you that pattern when he multiplies (a+b)^2 = (a+b)(a+b) = a^2+ab+ab+b^2
      Notice - the 2 middle terms match! They are like terms and combine into a^2+2ab+b^2
      If you square any binomial (a+b)^2, your result will be equivalent to a^2+2ab+b^2

      Sal applies this pattern when you squares (7x+10)^2.
      "a" = 7x
      "b" = 10
      So, using the pattern...
      a^2 = (7x)^2
      2ab = 2(7x)(10)
      b^2 = 10^2

      To better understand the 2(7x)(10), use FOIL. But, I'm going to do it without actually performing the full multiplication.
      (7x+10)(7x+10) = 7x(7x) + 7x(10) + 7x(10) + 10(10)
      7x(7x) is 7x^2, the same as a^2 using the pattern.
      7x(10)+7x(10) = 2(7x)(10), the same as 2ab in the pattern
      10(10) = 10^2, the same as in the pattern.

      Hope this helps.
      (3 votes)
  • old spice man green style avatar for user tdurnil000
    what if you have (a*b)^2. you would not use special cases right. also what would you do if it is (a/b)^2. what would you do then
    (2 votes)
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  • boggle blue style avatar for user Kalshi
    What's F.O.I.L?
    (2 votes)
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  • female robot grace style avatar for user Shaanae
    Where did the 2(7x)(10) come from wouldn't you do 7x*7x+7x*10+10*7x+10*10?
    (1 vote)
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Video transcript

We're asked to simply, or expand (7x + 10) ^ 2 Now the first thing I will show you is exactly what you should NOT do, well there's this huge temptation. A lot of people will look at this and say oh, that's (7x)^2 + 10^2. This is WRONG. And I'll write it in caps. This is WRONG! What your brain is doing is thinking if I had 7x times 10 and I squared that, this would be (7x)^2 times 10^2. We aren't multiplying here, we're adding 7x to 10. So you can't just square each of these terms. I just wanted to highlight, this is completely wrong, and to see why it's wrong, you have to remind yourself that (7x + 10)^2 is the exact same thing as (7x + 10)(7x + 10). That's what it means to square something. You're multiplying it by itself twice here. So this is what it is, so we're really just multiplying a binomial, or two binomials, they just happen to be the same one, and you could use F.O.I.L., you could use the distributive method, but this is actually a special case: when you're squaring a binomial, so let's just think about it as a special case first then we can apply whatever we learn to this. So we could've just done it straight here, but I want to learn the general case so you can apply it to any problem that you might see. If I have (a+b) squared We already realised that it's not a squared plus b squared That is a plus b times a plus b. and now we can use the distributive property We can distribute this a + b times this a So we get, we get a times a plus b and we can distribute the a plus b times this b plus b times a plus b, and we distribute this a we get a squared plus ab plus b times a is another ab And I'm just swapping the order so it's the same as this. plus b times b which is b squared. These are the same or these are like terms. So we can add them. One of something plus another of that something will give you two of that something. 2 ab. We have a squared plus 2 ab plus b squared. So the pattern here, the pattern here, if I have a plus b squared it's equal to a squared plus 2 times the product of these numbers plus b squared. So over here I have seven x plus ten squared So this is going to be equal to seven x squared seven x squared plus 2 times the product of seven x and 10. 2 times seven x times 10 plus 10 squared. So, the difference between the right answer and the wrong answer is that you have this middle term here that you might have forgotten about if you did it this way. And this comes out when you are multiplying all the different combinations of the terms here. if we simplify this, if we simplify seven x squared That's seven squared times x squared. So seven squared is 49 times x squared When you multiply this part out 2 times 7 times 10 which is 140 and then we have our x. No other x there. And then plus 10 squared. So plus 100. And we are done.