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## Special products of binomials

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# Squaring binomials of the form (ax+b)²

CCSS.Math:

## 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.