Everything You Need to Know About Expanding Double Brackets | GCSE Maths

Can you complete this GCSE exam question in under a minute, without any mistakes whatsoever? Well, if not, that's exactly what we're going to go through in this video. Now the reason why we want to start with expanding double brackets when we talk about quadratics is because it fits in really, really nicely with all of the other content that comes up later on. So here I've given you a little list of what comes up. So we've got expanding brackets, then we've got factorising quadratics, then quadratic expressions and quadratic graphs, which covers all of the GCSE content, which is a lot, as you can tell, and then it moves straight on into A-level maths. So this is something that, regardless of what level you're at, you need to be extremely good at and really kind of boost those fundamentals in order for us to get those top grades. So very quickly, I've got a little prior knowledge for you. I'm going to give you, let's say, 30 seconds, just have a go at these questions. Make sure that you can do them, probably 10 seconds per question, actually. Now pause the video and let's get started with that. So when you're doing these questions, you need to make sure that you have access to all of the different recommended knowledge. So for the first one, 3x plus 5x, why is that here? Because we're going to be collecting like terms. Now like terms have to have the same letter and the same power. In this case, 3x and 5x are both x's. So if I have three x's in one hand, five x's in the other, in total I have eight x's. You literally just add the numbers in front of the x's. The next one is just expanding a linear bracket, something that should hopefully be quite easy for you. So again, you're going to times everything on the outside of everything on the inside. So we have 4 times x, which gives me 4x. And then we have 4 times 3, which gives me 12. So we're going to put a plus 12 on the end there. Again, question three, we have, once again, simplifying expressions. Now here we have 7x minus 2x plus 4. The common mistake here is people adding the 4 on to the x as well. Remember, 4 is just 4. The other two expressions have x's in them. So if it doesn't have an x, it can't be added to 2x's. So here we have 7x take away 2x. So if you have 7x's and you take away 2, what are you left with? You're left with 5x, and that plus 4 just stays on the end. Because again, it's just 4. Multiplying negatives, again, something that can be quite tricky. Remember, if a negative and a positive times each other, or a positive and a negative, the answer is negative. So I'm going to put a negative sign there because it's a minus 3 and a positive 5. Then you just do 3 times 5, which is 15. You don't even really have to think about it. Now, if the sign is the same, so they're both negative or both positive, then the answer is positive. So I'm not going to put a negative on that answer line. And then we just have 4 times 6, which is, of course, 24. Hopefully that wasn't too bad. Now, here's the thing that we need to remember. Now, with expanding double brackets, a lot of people talk about FOIL, but don't use FOIL. It's ridiculous. It's stupid. It doesn't even work for all of the questions. So what is the point? Because the second they change the brackets, FOIL no longer works. It FOILs it, I guess you could say. So instead, all you need to think about is that every term multiplies every term. So I'm going to go through an example to explain what I mean by this, but I want you to remember this mantra. I want you to be saying it all the time. Every term multiplies every term. So here we have three examples. When I say every term, what do I mean? Well, as you can see in this first bracket here, we have two terms. We have x and we have 2. And then in the other bracket, we have x and 3. The key thing here is that both of those terms in the first bracket have to multiply the terms in the other bracket. So as an analogy, let's talk about handshakes. Let's say we have two people in the first bracket, two people in the second bracket. Each person has to shake the hands of everyone in the other bracket. So let's go through this. Here's how I do it. So I'm going to start with the x. And then we're going to have x multiplied by x, which gives me x squared. Then we have x multiplied by 3, which gives me 3x. And now we're going to move on to the 2, because that x has already multiplied everything else already. So I'm going to circle that. And we have 2 multiplied by x, which gives me 2x. And then we have 2 multiplied by 3, which gives me 6. So as you can see now, everything in the first bracket, each term in the first bracket has multiplied each term in the second bracket. This makes sense, right? Because the x is times the x and the 3 to give me x squared and 3x. And the 2 has multiplied the x and the 3 to give me 2x and 6. Now we're going to simplify. So if I have a look at this, x squared, are there any other x squareds? No, so that has to stay the same. So I'm going to tick that off. Now the 3x and the 2x, they are both x's, which means I can add them together. x squared and x is not the same thing. So it's just the x's are the same. So 3x plus 2x would give me 5x. And then that 6 is just by itself on the end. So we just have a plus 6. And that is it. You have now expanded and simplified. That's going to be two marks in a GCSE paper. I have created in the description a bronze level worksheet for you to practice exactly what we've just done. And to test yourself with work solutions is in the link below. We're going to move on to stuff that is far harder. That's coming up, though. So again, if you find this a bit easy, good. There's a silver and gold level that we're going to get onto, which takes us on to exam questions. And then also kind of beyond exam questions like grade eight and nine stuff, which is very interesting. So let's try the exact same thing again. Let's start with blue. So again, we'll start with the x. I'm going to do x times x, which gives me x squared. x times five, which gives me 5x. And then we're going to move on to the one which I'll use the green colour for. One times x, which is just 1x. You can just write x if you'd like, which is probably a little better. And one times five, which is five. And again, what's the next step? Simplify x squared. There's no other x squared. So it's going to leave as x squared. The 5x and the 1x are both x's. We can add them together to get 6x. And the five on the end is a five. Shockingly, I know. Cool. There you go. Right. How quick was that? So let's keep going again. Try one for yourself. If you have a piece of paper or try and do in your head, maybe because I would recommend it. Now we have negatives a little bit harder. We have x times x, x squared. x times four gives me the 4x. Then we have minus two. So remember, the sign does matter. So that's minus two times x, which is going to be minus 2x. And now minus two times four. Now, hopefully, you can see why we did that prior knowledge is going to be minus eight. And again, we're going to simplify x squared. There's no other x squared. Then we have 4x. Take away 2x. So it's not going to be 6x. It's just going to be 2x. So I'm going to put plus 2x. And then the minus eight on the end, like so. Cool. Hopefully good so far. Let's do a few more. Again, feel free to pause the video and have a look. This is getting slightly harder into kind of the silver territory. But again, if you still find this easy, jump ahead to the gold stuff, because some of that is really, really interesting. It hasn't even come up in exams before some of it, which means it could come up this year or next year. So again, let's work through it. Hopefully you've had some time to pause the video. Here we have the 2x. 2x times x. So the two stays the same because we have there's no number in front of the x, just one. x times x is x squared. Perfect. We have 2x times three. So we're going to do this three lots of 2x, right? So 2x plus 2x plus 2x. Or in other words, three times two is 6x. Then we've got the one. One times x is going to just be x. I'm not going to write the 1x. Hopefully that's okay. And then we have one times three, which gives me three, of course. And then again, gathering the like terms. Well, 2x squared. There's no other x squared. So don't have to do anything there. But here we have 6x plus x, which is going to be 7x. And then we have the three on the end again. Cool. Now we're going to have numbers in front of x and negatives. It's making it slightly harder. Perfect. So we have a 3x here times x. So what does that happen again? What does that become? 3x squared. Beautiful. Then we have 3x times 5. Again, have a think. Well, again, you're just multiplying the numbers, right? So 3 times 5 is 15. So we have 15x. So that's that done. Why is it done now? Because again, the 3x is times both of the things in the other brackets. There could be three or four things in the other bracket. But in this case, there's two. Now we have a minus 2 times x, which is, of course, minus 2x. And then minus 2 times 5, which is minus 10. Gathering that together, x squared. How many x squareds are there? There's just three. There's no other term. So we're going to do 3x squared. 15x does have a minus 2x. So 15 minus 2 is 13. And then the minus 10 on the end. Cool. Next one. Again, 4x times x gives me 4x squared. 4x times negative 2. Remember, positive and a negative makes a negative. So 4 times minus 2. I know it's going to be a negative number. Then 4 times 2 is 8. Perfect. Then 1 times x is just x. 1 times minus 2 is minus 2. Have you noticed I'm stopping the whole color coding thing to see if it still makes sense to you? Then we just have 4x squared, because once again, there's no other x squared. You might be noticing a pattern with that. I hope you are. Minus 8 plus 1 is minus 7x. And then minus 2. So, last two questions. Again, do pause the video. Make sure you're having a go, because the more you practice, the easier this becomes and the fluency increases. Because again, it's in all of your GCSE exams. But also, if you want to do A-level maths, this is something you have to just be lightning fast at. Let's go through it. 2x times the x gives me 2x squared. 2x times negative 4. Well, because it's a positive and a negative, I know the answer is negative. Then 2 times 4 is 8, so it's going to be 8x. Then we have minus 3 times x, which gives me minus 3x. And then minus 3 times minus 4. Well, remember, they're both negative, which means the answer is positive, which gives me positive 12. And again, we're going to gather the like terms. So, once again, there's no other squared, so we're just going to leave it as 2x squared. Minus 8x minus 3x. Well, if it's minus 8 degrees and it goes 3 degrees lower, that's going to become minus 11x. And then the plus 12 on the end once again. And for the last question, before we move on to something slightly more challenging, we have 5x times 2x. Well, 5 times 2 is 10. x times x is x squared. 5x times 3 is going to give me 15x. 2 times 2x is going to give me 4x. 2 times 3 is going to give me 6. Just so you know, there is a shortcut with this. We can do it a lot faster, but I'm going to let you come up with it. When it comes to factorizing, that shortcut will become a lot more obvious, which I'm hoping you guys will actually spot because it will make your life a lot easier. That gives me 10x squared. 15x plus 4x is 19x. And then we have the plus 6 on the end. And that brings us to the end of that section. So once again, before we move on, there is a bronze worksheet in the description, which I would recommend going through. There's also a silver, which you should be able to do now. After this, we're going to get on to some gold stuff. I'm only going to do one though. The reason for that is the gold stuff has a lot of problem solving, which I want you to work through. There are work solutions already on there as well, which you can use to help you. But let's move on to the next section. The next type of question that they love to give you in GCSE, and in A level as well, is spotting mistakes. So what I'm going to do is I'm going to do these questions, and I'm going to make a common mistake. And I want you to try and spot it. And then I'm going to tell you what it is, of course. But again, I want you to try and spot it first. So I'm going to do this in red because obviously we're doing it incorrectly. So once again, we're going to have x times x, which gives me x squared. You're going to have x times 3, which is 3x. And you're going to have 2 times 3, which gives me 6 as my final answer. Right? Right? That's the final answer, yeah? Okay, so what did I do wrong? Well, the thing I did wrong is that I missed out one of the multiplications. I didn't do 2 times x, which would give me a plus 2x, which means my final answer would be x squared plus 5x plus 6, which is obviously not what I got because I skipped that one out. So again, that's a really, really common mistake. I know you might spot that and think, well, that's really easy. But again, if you're in an exam, you're under pressure, very easy to make these mistakes. So next one, let's see what I do wrong here. So again, we have x times x, which is x squared. x times minus 2, which is minus 2x. Minus 4 times x, which is minus 4x. And then 4 times minus 2, which is minus 8. So I'm going to gather the like terms. Now we have x squared minus 6x minus 8. I wonder if you can spot where I went wrong there. I'm also changing the way I'm wording this to see if you can actually spot it. So I'm saying the wrong thing as well in some cases. This should be long enough. So if I was doing this in an exam and I saw this question, because they can give you something that looks like this and say spot the mistake and write the correct solution. What I would do is I would do it myself. So we have x times x, which gives me x squared, which I'm fine with. Then we have x times minus 2, which does give me minus 2x, which I'm fine with. Minus 4 times x is minus 4x, which I'm fine with. So so far, these first two terms are 100% correct. So that means the mistake probably is minus 4 times minus 2 is not minus 8. It should be plus 8. So those negatives really do trip up a lot more students than you might think. But is what it is, right? It happens to people. OK, the next one I hope would be obvious. But, you know, it's going to be less obvious when we do expanding triple brackets, put it that way. So again, we're going to have x times x, x squared. x times 2 is 2x. 1 times x is x. And then 1 times 2 is 2. And then that in total, well, there's 1, 2, 3, 4. So there's 4 x's plus 2 as my final answer. Again, I did say that you probably spot this one slightly faster than than the others. The problem is, is I collected unlike terms. So here I added the x squared to the 2x and the x, which is, of course, not correct. The answer should be x squared plus 3x plus 2. Because, again, where did that x squared go? The other time I've seen this is some people do x squared plus 2x is 3x squared and they get 3x squared plus x plus 2, which looks correct because there's x squared x and then a number. But it's not correct. So, oops, right? Oops, it happens. Here's the next really common mistake. Again, hopefully you're going to laugh at this one because you've, you know, you've watched and you've learned something in this lesson. But I see this, which is 2x times 3x, which is 6x squared. And then 1 times 4 is 4. And they're done. So I'll give you a second. Hopefully it's fairly, fairly obvious. Yeah, the problem here is that people don't multiply everything in the first bracket by everything in the last bracket. And this is where foil also foils them, right? Because some people forget about foil and all kinds of weird stuff happens. It also happens when the brackets themselves are a bit weird. So, for example, they could do something like this, 1 plus 2x, and then 2x plus 1 or something like that. Where again, people doing first, outside, inside, last, foil, they get mixed up with the terms. Whereas instead, in my way, it's just I'm multiplying everything in the first bracket by everything in the second bracket. There's no need to worry about that. Now, I'm going to go through some diagnostic challenge questions. Now, there's three of them, and they progressively get a bit harder. And the last one is definitely something quite juicy, which you see all the time in A level. It has appeared once in GCSE, kind of. And in the gold worksheet, I have a ton more examples of it, which is going to be really, really fun for you. So let's have a look. Here are three challenges. You can't really see the third one actually yet. But let's start the challenges, shall we? Again, I'd recommend you pause the video and then watch my solutions afterwards. And we can compare and see what we got right and what we got wrong. So the first question is a classic spot the mistake. Happens all of the time. So how would I go about doing this? Well, I mentioned this earlier, which is I would do the question myself. So literally on the side, I'm going to expand these brackets. So x times x is x squared. x times 2 is 2x. 4 times x is 4x. 4 times 2 is 8. And gathering the like terms, we have x squared plus 8x plus 8. So what are the mistakes? Well, firstly, if I look at this plus 2, which is the thing that was sticking out to me, I got plus 8, he got plus 2. Why is that? Well, that means he didn't do, he did not, or she, I guess, multiply 4 by 2. Instead, it looks like he just put the 2 on the end, which if you remember, in the previous example here, when I did one of those common mistakes, happens a lot, right? People forget to multiply one of the numbers, happens a lot. So, okay, what's the other mistake, which I'll do in blue, why not? Well, what else is different? Well, he got 4x and we got 8x. So how would that have happened? But it looks like he's done 4 times x is 4x, but he didn't do x times 2, which again, happens more than you might think. So that would be x by 2, which means obviously his term is going to be incorrect, right? Okay, the next type of question that does come up, again, this is a lot more frequent in A level, I'm not going to lie to you. But again, there's no reason why it can't come up in GCSE. And it does come up in certain examples a lot more as well. So it says here, which expression is identically equal to 4x squared minus 25? And it gives you a bunch of examples. Now, honestly, the surefire way to do this is just to expand all four of these brackets, all of them. But can we be a bit smarter? Because if you're good at maths, you're lazy, right? You want to find the easy way to do it. Well, if I compare the numbers at the beginning, is it any that kind of stick out to me as not being correct? Well, to be honest, it could be any of them, which is a bit of a shame. But you might notice this from when we do factorising, but that looks like a difference of two squares. So if I have a look at all of the different values, if I look at b, and I expand b, I get 4x squared because of 2x times 2x. 2x times 5 is plus 10x. Minus 5 times 2x is minus 10x. And minus 5 times 5 is minus 25. These two add up to zero, so we just get 4x squared minus 25. And this is the key reason why b is correct. It's correct because the constants at the end give us the exact same number, which means we get rid of that middle term. That x term, which is zero in this case, disappearing only happens when the coefficients of x are the same, and the numbers inside the brackets are plus or minus the same number. So as another example, if I had 3x minus 4, 3x plus 4, I will get no x term in the answer because the plus 4 and the minus 4 cancel out with the 3 and the 3. Now, I've got a much harder question on this exact thing here. I have a much harder question in this gold worksheet. If you find this interesting, I'm really looking forward to you trying that question. Again, the link to that is in the description. It's completely free, no worries about it. The coefficient of x, and I'm going to zoom in as well, the coefficient of x in this expansion here is 1. We need to find a. OK, cool. So what would I do? Well, I would expand the brackets, right? Because, I mean, it's expanding double brackets the lesson, so let's see. We have 3x times 2x is 6x squared. Then we have 3x times minus 5, which is minus 15x. We have a times 2x, which is 2ax. Then we have a times minus 5 is minus 5a. And it says the coefficient of x is 1. So what's a coefficient? Well, the coefficient is literally just the number in front of an x. Well, in front of anything, really. I could say the coefficient of x squared. I could say the coefficient of a, coefficient of b, etc. This is the number in front. So what they're telling me is that the number in front of, the number in front of x. So this minus 15 and this plus 2a should equal 1. So in other words, I know that minus 15 plus 2a equals 1. If you don't like that, the other thing you could do is you could say that minus 15x plus 2ax equals 1x. And then just divide the whole thing by x. We have minus 15 plus 2a equals 1, which is the exact same thing, right? So either one of those two gives you the same answer because it's literally the same working out. But the idea is if we use the word coefficient, we can be more efficient. Sure, we can be a little bit more tidy with our work. Coefficient just means the number in front. So they're saying the number in front of x should be 1. But what's the number in front of x so far? It's minus 15 and plus 2a. So now what I can do is I can add 15 to both sides. So I get 2a equals 16. And I can divide both sides by 2 to get the answer of a is equal to 8. Looks like we ate with that question. So I promise you I will stop at some point. So all of that is to say, you know, congratulations, you've managed to do it. But I would recommend that you go onto that link in the description. It has a bronze, silver, gold worksheet. So you can practice the relevant skills that you might want. And then afterwards it has a mastery quiz. The mastery quiz, if you can do that without using any notes at all, means you've really secured this topic and you're ready for that grade 8 and 9. Anything you get wrong, you can go back to the worksheet or back to the video and retry it and get it all correct, of course. But again, what I want you to do is after you complete it, reply with your confidence score out of 8. It explains everything on the quiz, don't worry. Other than that, I will see you in the next lesson, which is Quadratics 2.