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### Course: Class 9 (Old)>Unit 2

Lesson 6: Standard identities

# 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.

## Want to join the conversation?

• I don't get it. How is (7x)^2 + (10)^2 wrong?
• 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.
• why is 7x squared+ 10 squared wrong? I don't get it.
• 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.
• At isn't pretty redundant to point this out this late in the course wouldn't it help to emphasize this point earlier?
• True. However, perhaps some people might forget.
• Does this apply to numbers as well, lets say (2+8)^2, or is it only when there are variables involved?
• 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.
• 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.
• 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.
• would this work for (-5wx^5)^3? how would I do it? thanks
• 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 :-)
• 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!
• 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.
• this was in an exercise before this vid no wonder i didn't understand it at first