Fill in the gap: ________ is the opposite of expanding.

Factorise x² + 11x + 24.

Factorise x² + 14x + 24.

Factorise x² + 8x + 16.

Factorise x² - 25.

A quadratic expression  x² + bx + 20 can be factorised. Find all the possible values for b when b is positive.

Factorising Quadratics 1 Quiz

﻿Hello, my name is Mrs. Buckmire, and today, I'm going to be teaching you about factorising quadratics. So it's a bit more than if you were in factorising quadratics one, this is taking that learning a bit further, okay? So first, make sure you have a pen and paper. Now, please do pause the video, whenever I ask you to, so if I want you to think about something or have a go, please do pause it, but also whenever you need to, so if you need more time, take your time. And sometimes it is useful to actually go back and rewind the video. You don't need to just watch it all in one. Rewind it, see little bits again, and that can be useful for your learning. So just go at your pace basically. Okay, let's begin. So for the try this, how many different quadratics can you make by arranging the cards? Give the expanded form of each one. So, you're going to put those cards in. So you've got the four cards on the right hand side here, and see what different questions, then expand them. Pause video in three, two, one. Okay, so here below, you've got the four different answers that I think, well the four different questions I think you could have created. So for the answers, the first one, what did you get? X takeaway two, times X takeaway one. I got, X squared, takeaway three, X plus two. The next one, I got X squared, takeaway four, X plus four. You might see actually you could also write that, how else could I write to be more efficient? If I want to write it using my squared, oh, this isn't working. And here we go, it could also be written as, X takeaway two squared, and here we have, X squared, takeaway six X, plus eight, and X squared, takeaway five, X plus four. So well done if you've got those. So Binh has developed a strategy for factorising quadratics. So X squared, plus eight X, plus 12, equals X plus two, plus X plus six, that's the answer. But she said, I found all the factors of 12 and then picked the two that sum to give the coefficient of X. Okay, coefficient, what does that mean? Good, coefficient X, is the number in front of X, so here it would be positive eight. So, a coefficient X, that's the number in front of X. Where can I write that, let's put it up here. So, maybe write yourself a note if you didn't know that. So I want you to explain how the strategy works. So it could be helpful to do a diagram, to maybe draw the array. Otherwise, just think about it. How did the strategy work when factorising the expression below? So X squared, plus nine X, plus 20. So don't spend too much time on this, but do have a think about how it works. Maybe look back at the examples from the try this and see, okay, what's going on? What's going on that's got to do with the coefficient of X? Pause the video in three, two, one. Okay, so if we did it an array, it would look something like this. So where we know that, for example, this, let me doraw with black. This is X squared, this is my X, and my X, and this is my a and b, but I don't yet know what a and b are. I know they multiply to give 12. So that's where the factors of 12 come in, because we know that a times b, this is a, and this is b, so that area is 12. So let's see, factors of 12, one and 12, two and six, three and four. To add to give the coefficient of X, why does that work? So here, we would have chosen two and six. Cause two plus six equals eight, and two times six equals 12. So let's say, this was two, and this we know now is six. Ah, because actually here, it will be six times X. So this area is going to be six X. And this one is two times X, so this area is two X. Six X, plus two X, equals the coefficient of, equals eight X, so equals actually this second term here. So actually, the factors, whichever two add up together to equal the coefficient of X, that's how we can work it out as well. Okay, so we find the factors of the number, and then we pick the two of that sum to get the coefficient of X. Okay, maybe it's a good idea to write that down. That seems like valuable knowledge there. Let's look at how that strategy could work, when factorising X squared, plus nine X, plus 20. Okay, so we want multiply to give 20, and what does it have to add to equal? the coefficient of X, what's that? Okay, so add to equal nine. Okay, let me just write out the factors of 20. So, one and 20, two and 10, four and five. Oh, four plus five, equals nine, so therefore, it's going to be X plus four, X plus five. That seems correct, well done. Okay, but what about these? These are negatives, I wonder if it still works. Let's try. So, they're going to multiply to give what? What's the number? So negative four. And add to give? Three. Hm, so if it has to multiply to give negative, would then I need negative or positive, or positive and negative? So here, is where when you think really carefully about our factors. So we could have, negative four times one, or four times negative one. We could have, two times negative two, or negative two times two. But that's the same. So these are our options. And now add to give three. Negative four plus one equals the negative three. So that one's not right. And four plus negative one, oh, that equals three, doesn't it? Yes, so therefore, it must be when factorised, X, plus four, times X, take away, one. Now, what could I do to check if I wasn't really sure? Excellent, I could expand it. Pause the video and expand it now. Good, did you get the same answer? Yeah, it seems correct, nice. Right, what about this next one? Let's have a little look. So, we add to give, sorry, first let's do multiply. Cause I think it's easy to start with that. Multiply to give. Negative four, and add to give. Negative three. Okay, we've already did the factors, didn't we? So, I actually know the factor pairs already, I can copy it. My negative four and one, we had four and negative one, and we had two and negative two. Which one adds to give negative three? Good, negative four plus one. So this one equals to X, take away four, and X plus one. Okay, make sure you're confident. That maybe even listen to that again, or pause it and write it down. And then let's have, let's connect to actually how it could work with in array. Okay, so we have X squared, takeaway six X, plus eight. So, the X squared part, that's fine, but now plus eight. So we add on eight that's on the outside, and this is where it goes a bit weird. So the fact that we have negative six X, so our X values are now going to be negative. So it's like we're taking away, but then actually here we'd be adding them on. So, this is where the just kind of falls through here, but we can still use this as a tool, so it's still super useful. And if you use an , then actually I know, that you can flip over to the negatives, so that kind of makes sense. So kind of imagine that for a and b here where flipping it over those bits to the negative bits. But otherwise just think of it as a tool, this will still help us out. So, we're still thinking about two numbers, for a and b, that multiply together to equal eight, but add together to equal the coefficient X, which is negative six. So we can write this down. So, we have two numbers, so multiply to give eight, but add to give negative six X. So I think that they're going to have to be negative. So we could have like, negative eight and negative one. Or we could have negative four, and negative two. So, we could write it as X takeaway, eight, times X, takeaway one. Could be one realisation, or maybe it's X takeaway four, X takeaway two. So if you'd like, you can pause that and actually decide between them. But using Bhin's rule, we know that when we add them together, they need equal what? Yeah, negative six. So when we add negative eight and negative one, we get negative nine. We add negative four and negative two, we get negative six. So that's how I know actually it is this one, and this is the correct answer. So we'd have X takeaway four, X takeaway two. So here, we can use this tool to kind of work that out as well. So, factorise the expressions to form an answer from the box, which questions can you not factories? Which four answers do not match any of the questions? So here are your questions. There might be one, maybe two, that you cannot factorise, okay? So as you'll think, ah, I'm trying, but it's not working out, okay? And here are all the possible answers, and there's four that do not match any of the answers any of the questions, sorry, at all okay? But do try and factorise, don't go and just expand them all, and see which ones. Factorise is what I want you to practise okay? And the answer there, just give a bit of confidence, like oh yes, I've got it right, it is there. It's not there and it's probably wrong, okay? So pause the video and have a go in three, two, one. Okay, so let's do, let's just look at the first one. So the first one, we have negative 12. So the two numbers that multiply to give negative 12, and add to give positive one. So the coefficient X here is one. If there's no number, and it's plus, it's plus one. If it was negative x, what would the coefficient be? Good, negative one. So two numbers or more, applied to give negative 12, but then when we add them, they're going to be a positive. So, I kind of know that actually I need my positive number to be bigger. So it could be like six, and negative two. It could be like, 12 and negative one. It could be like, four and negative three. So, I've been a bit strategic there. So, it's fine if you have to write them all out. So actually, yeah, four plus negative three, equals one. So that's why it matches with this one. So what I've done, I've matched them by colour, so, that one matched with that one, this one, to this one. For some of them the colours aren't coming up. This one matched to this one. This one matched to this one. And this one, matched to this one. This one didn't match to any. Because actually, if you wrote out the facts of negative 20, none of them add up to equal to nine, okay? So, we can't factorise everything yet, okay? But, yes, so there are no factors here that work out. So this one was the question you cannot factorise. And these were the four answers that did not match any of the questions. So some of them were kind of answers to try and make you make a mistakes, so hopefully you didn't. Do pause and check which ones you've got correct. And maybe just check again the ones you didn't. Cause, really all of, you should get all of them correct. Because, you could check them by expanding. So after you actually worked them out, then do expand and check oh, did I get it correct? Cause you should be really confident with your expanding. Okay? Well done. Okay, for our explore, Does Binh's strategy always work to factories quadratics? So. This question is what she used. She said, I found all the facts of 12 and then picked the two that sum to give the coefficient X. So we know this. For the below quadratics when doesn't it work? Can you explain why not? So here they are, if it doesn't work, can you maybe figure out a way that you could factorise it? I've not tricked you here. You can factorise the ones, all the ones that are here. But, there's, you have to kind of to use a different method, use your imagination a bit, okay? So don't worry if you can't quite factorise the ones where it doesn't work, but try I'm factorised one way it does work, and try and, figure out which ones it doesn't work for, okay? Pause the video in three, two, one. Okay, so I've changed, Bihn's, sentences here to I found all the factor pairs, and then pick the two that sum to give the coefficient of X, okay? So it's just more generalised, so it works for different ones. So, when doesn't it work, can you explain why not? So these are the two that it does work for. Can you factorise those? Pause it now and do it now if you haven't. Good, it would be X takeaway 11, X takeaway two. So, negative two, times negative 11 is plus 22. And negative two, plus negative, negative 11, plus negative two, equals negative 13, okay? And then negative 10, times negative 11, equals, positive 110. And negative 10, plus negative 11, equals negative 21. Okay, so which ones didn't work? All the other ones. how to work them out. So, do you notice how with this first one, all of the numbers can be divided by two? So we actually could factorise two out, to get that. Now, this looks like something maybe you worked out before, maybe you can spot, you could work out the factorization of, yes? Have a go if you haven't done it. Two numbers that multiply to give six and add to give five? Three and two, so it's two, times X, plus three, times X plus two. Nice, maybe if you're like, yeah, I get that, try the other two. Okay, we can take two out again. This time it's got negative involved, so two numbers that multiply to give negative six and to give five. Positive six and negative one? Yeah. And final one. What do we divide by this time? We need the coefficient X squared, to equal? Good, you know it, one. So here we have to divide by six. And now actually, this inside brackets, is the same, as this one. So actually, it's six, lots of X plus six, X takeaway one. Really, really well done, if you got that. So the, when this, when Bihn's strategy works, is when the coefficient of X squared is one. So each time we're making the, the number in front of X squared just being one, which means, you know, in that math superficial, we don't have the number, so it's just X squared. And that's when we combined all the factor pairs that, that form the number, and then pick the two that sum to give the coefficient of X. Really, really well done if you got that. Thank you so much for all your hard work today. I hope you understood that. Do go back and look over at any bits that were tricky. The main thing is for you to be able to factorise those quadratics, so those first two there, the ones I want you to be able to do. Do have a go at exit quiz, really good opportunity to show off what you've learned and also get some feedback and just see if there's any gaps in your knowledge then, or understanding, that hopefully with the feedback I can help you to close them. Have a really lovely day, and hopefully I'll see you in another lesson. Bye.

Have a go at the questions below. Make sure you read the questions very carefully! Good luck!

Cala says that you cannot factorise x² -9x + 8 because there does not exist a pair of factors for 8 that add to give a negative number. Is she correct?

Factorise x² - 3x - 10.

Factorise x² - 9x + 14.

Factorise x² - 36.

A quadratic expression  x² + bx - 20 can be factorised. Find all the possible values for b when b is negative.

Factorising Quadratics 2 Quiz

In this lesson, we will further develop your ability to factorise quadratics by spotting factors of terms.

Maths

Y9 Maths

Key Stage 3

Year 9

Factorising quadratics (Part 2)

Quadratic expressions (9.9)

Algebra

THEME

UNIT

Recognising linear and non-linear graphs

Plotting quadratics (Part 1)

Plotting quadratics (Part 2)

Quadratic contexts

Maximum and minimum area

Distributive property

Expanding double brackets

Difference of two squares

Factorising quadratics (Part 1)

Sketching quadratics

Interesting quadratic patterns

Match the expression equivalent to (x+4)(x-4).

Which of these tables displaying coordinates of graphs is non-linear?

Complete the tables below and plot the graphs. Select a co-ordinate the graphs have in common.

A square is cut out of the rectangle. What is the maximum area?

Antoni says this is not a quadratic as it does not go through the x axis. Do you agree?

Cala says (x+2)(x+4) is the same as (x+4)(2+x). Is she correct?

Expand and simplify (x+2)(x+5)

Select the words that best fill in the gaps in order: The __________ is where the curve crosses the y axis, when _______.

Which graph could be a sketch of -x² + 3x - 2? *Hint* Try out some coordinates to check!

I pick two consecutive integers. I square the bigger one and then subtract 4 of the smaller one. Zaki writes a different algebraic expression to me but is still correct. Which one could he have written?

Lesson overview:Factorising quadratics (Part 2)

Core Content

Quadratic expressions (9.9):