Lesson video

Hello, I'm Mr. Coward, and welcome to today's lesson on rearranging quadratic equations.

For today's lesson, all you'll need is a pen and paper or something to write on and with.

If you can please take a moment to clear away any distractions, including turning off any notifications, that would be great.

And if you can, please try find a quiet space to work where you won't be disturbed.

Okay.

When you're ready, let's begin.

Okay.

So time for the Try this task.

How many different ways can you write this equation? 2x plus 3y equals 5z.

So I've given you a bar model and two examples just to get the ideas flowing.

So pause the video and have a go.

Pause in three, two, one.

Okay.

Welcome back.

Now, here are some of my ideas.

Well, in fact, before I give you some of my ideas, can you see where this comes from? So here we've got 5z subtract 2x equals 3y.

Here we've got 2x plus 3y divided by five would just be z.

Okay, so what else could we have? Well, could have 2x equals 5z subtract 3y.

So, well, if that's true, then we could have, x would just be half of that length.

So that length divided by two.

What would y be on its own? Well, y would be 5z minus 2x divided by three? Hmm.

What else could we have? Well, what is? What about 2x plus y equals 5z minus 2y? What about that? What about this? 5z minus 3y minus 2x equals zero.

We could have that as well.

There's so many different ones we could have.

So these five are just to start.

So if you struggled a little bit, maybe see if you can find one or two more now, okay, using this kind of idea or this kind of idea or any of these kinds of ideas, okay? So if you didn't, if you found it quite difficult, have another go.

Otherwise, let's move on.

Now I did this equals zero one for a very important reason.

'Cause that's going to be something we're focusing on now.

So can you make all the following equal to zero? So what do I mean by that? Well, say I had x plus seven equals 15.

I want to make that equal to zero.

So I want to make that side equal to zero.

So that equal to zero, sorry.

So I take off 15.

So I get, x minus 8 equals zero, okay.

So you need to do the same on here to try and get one side equal to zero.

And I think I would make the right-hand side equal to zero for all of these.

Okay.

So pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Let's see what we've done.

So I would take 10 from both sides and I would get x plus two equals zero.

I would take 10 from both sides.

I'll get x minus 22 equals zero.

I would take 10 from both sides and I would get 2x minus 10 equals zero.

I would take y from both sides.

Okay.

Take 3y from both sides.

What to do here with this 3y? I would add 3y to get rid of that negative 3y, I would add 3y and then I would subtract 10.

So I get that.

Then I would add 2y here.

So I would get 2x plus 7y.

5y plus 2y is 7y, and I would take away 10.

Okay.

Hopefully you got a few of them right.

They got a little bit tricky towards the end.

So really well done if you did! Now, the reason why we've done this is because it's very important to be able to get a quadratic in this form.

Okay.

This is like the standard form for quadratics.

Well a is called the coefficient of x squared.

So it's just the number that goes with x squared.

b is called the coefficient of x.

And it's just the number that goes with x and c is just the constant.

So it's useful to get quadratics in this form.

Now it's useful.

We saw the last time when we were doing graphs, that when it equals zero, we can see where it crosses y equals zero or the x-axis.

So that's one of the reasons.

But another reason is to do with solving quadratics through algebraic methods, which we'll look at in the coming lessons.

So for each quadratic, find the value of a, b and c.

So if it's in this form, what would be the value of a? You may need to pause the video and have a think.

Okay.

So a would be five.

It's just the number that goes with the x squared.

Okay.

What would b be? 11.

And c? Two.

Ooh, what about here? This is a bit different.

Scarce, this one, isn't that? Hmm.

What would a be? We've got a secret one.

We don't really write that one, but we have got a one there.

It would not be zero, it would be one.

But b, our coefficient of x, okay? And we use that word coefficient a lot.

So maybe get used to it.

The coefficient of x would be zero because we have no x term.

And what would the constant be? Two.

Oh, well, there's quite a lot different about this one, but what's really new is that we've got lots of negatives here.

How's that going to affect things? Well, the coefficient of x squared is negative one.

What's the coefficient of x? Negative 3/4.

And what's the constant? Negative four.

Okay.

And you can kind of view it as that, because it's a plus here, right? So if it was, if we had say we had our 3x squared plus 5x minus two, okay.

Well then that's like thinking we've got plus negative two.

They're the same as each other.

Okay.

That and that is the same.

So because it's a plus here, that's why it's a negative because we want a plus to be there.

So that would mean that it's negative two, the constant is negative two in that case.

And in this case, it's negative four because we want to know what we're adding.

And because we're subtracting, we can think of that instead as we're adding on negative four.

Write the following equations in the form axe squared plus bx plus c.

Why is that not in the form axe squared plus bx plus c? Well, it's in a different order.

So we're just going to rewrite this.

Okay.

So we want our x squared first.

Then we've got 11 x, then we've got minus two at the end.

Okay.

And we can swap the order like that.

And you can swap it, especially if you thinking, because if you think that that is adding on negative two, rather than minus two, then addition is, can we say it so we can just swap the order? So we've just swapped the order here.

And that does not change our equation, okay? So swapping the order does not change the equation.

Oh, I should've gone down.

Oh, no.

So let me rewrite this.

That is 5x squared plus 11x minus two equals zero.

So I'd like you to pause the video and rearrange this equation.

So pause in three, two, one.

Okay.

Welcome back.

Now, hopefully you've rearranged it like this.

Okay.

It's a negative 2x squared.

'Cause it's like we're adding on negative two lots of x squared.

So we can swap the order around so we get our negative two at the start.

Okay.

Really well done if you got that correct.

Okay.

So write the following in the form axe squared plus bx plus c.

What's different about this one? This needs to be equal to zero, okay? It needs to be.

It's 10 you want in that form.

These are not equal to zero, but we can make them equal to zero.

So what would I have to do to make that, one of the sides, equal to zero? Well, I could take everything off that side, but that would, that would be a bit of a faff.

So what I'm going to do instead is I'm just going to take off that 3x squared there.

Okay, so it doesn't matter which side we take things off.

So it's just whichever is kind of easiest.

And I'll talk a bit more about that shortly.

Okay, how can I make this equal to zero? What can I add on x? So I added on x to both sides and there we are.

Okay, c, what would I have to do to make that equal to zero? Would I have to take two? So we get 11x plus 5x squared plus nine equals zero, but then we want to change our orders to get in the form x squared plus bx plus c equal to zero.

Okay, and the reason why is because it's easier when we get to factorise in a bit in that form.

There's nothing magical about it, but it just makes things easier.

And it's the order that we tend to write quadratics in.

And that's particularly important if you get to higher power polynomials.

So when you get to like fourth powers quartics or et cetera.

So it's kind of, it's nice to have them, starting from the highest power going to the lowest power.

Okay.

So let's rearrange this one.

Okay.

Well, I can add on 3x to both sides.

Add on 3x to both sides and I'll take two from both sides.

Okay.

So then we get equals to zero.

Now again, I could have brought everything over to that other side and just swapped it round.

So I would have had zero equals something and that's fine.

That's completely fine to do it like that, okay? And we'll talk more about that soon.

So I would like you to pause video and have a go at these six questions.

So pause the video and have a go in three, two, one.

Okay.

Here are my answers.

Hopefully you took x squared from both sides.

So you got, in fact, I've got my answers here.

You've took 3x squared from both sides.

So you got 3x squared plus 11x plus, definitely not plus two, definitely plus seven.

Okay.

You didn't see that.

Okay.

What about this one? Well, we added on 5x to both sides.

So we got 4x squared plus 6x plus seven.

We added on 5x to both sides.

So we got 5x, 4x squared, plus 5x plus seven.

We added on, we took away 3x squared.

We took away 3x and we took way two.

So that's quite a big one, that one.

So what did we do? We took away 3x squared.

We took away 3x and we took away two.

And if you want to do it in three separate steps, that is fine.

However, I was just doing it in one step, just for the sake of time.

And if you can do it in one step, that's awesome as well.

We just don't want to make a mistake.

Okay.

Now next one.

Hmm.

I'm going to discuss these on the next slide.

Okay, so e.

Well, you could have done this.

Let's take away 7x squared.

Let's take away 4x and let's take away two to get that side equal to zero.

So on this side, I would have negative 2x squared.

We took out 4x, so positive 7x and zero.

Okay.

Now what about this one? Well, I could take away 5x squared.

I could take away 11x and I could take away two.

So what would I get on this side? Sorry.

I should have wrote my operations in over here.

I do apologise for that.

We would have got 2x squared minus 7x equals zero.

Okay.

What is the same and different here? What is the same about e and f? Did you do what I did? Did you get them off in this form and you might have done.

Which do we prefer? Because this is the same question.

It's just that the things on the side of the equal sign is swapped around, okay? But it's the same question.

So we've got up 5x squared plus 11x plus two equals 7x squared plus 4x plus two.

And we got the same on the other side, but just the things on equal sign is swapped around.

So which one's correct? Well, they're both correct, but we tend to like a to be positive.

Because it's, not always, but most of the time we do, because it's just easier.

So it's easier if the coefficient of the highest power is positive.

So we prefer this.

So what would we have done if we'd got this and we didn't get this and we wanted it to be positive? Well, instead of doing that, you could have took 5x from both sides.

You could have took 11x from both sides and you could have took 2x from both sides.

So you would have had zero on this side.

So you would have had on that side, I'm going to write it over here.

You would have had zero equals 2x squared plus 7x.

And that is, you know, swap it round and then that's in that form, okay? You just swap those two round, okay.

Because you can do that because three plus two equals five, five equals three plus two.

It's just how you say it.

Zero is equal to 2x squared plus 7x.

2x squared plus 7x is equal to zero.

We can swap those order of the expressions around and my equation will still hold true.

Okay, but we could have also done this.

So say you got negative 2x squared plus 7x equals zero.

And you wanted to make it positive.

Well, you could times it by a negative one on both sides.

And that swaps all the signs.

So we get 2x squared minus 7x equals well, zero times negative one is just zero still.

So you can see now how we have the same thing there.

Okay.

So we could have took the 5x squared and the 11x and the two away, or we could have done what we did and times it by negative one to get a positive there.

And we get the answer either way.

Okay.

All of the following are quadratic equations.

Can you see why? So you may need to pause the video and have a think.

Okay, they are all quadratic because in every single case, we're multiplying an x by an x, an x by an x, and an x by an x, which gives us a squared.

So let's write them in quadratic form.

Let's write them in the form axe squared plus bx plus c.

So let's square this bracket.

Collect like terms and not square the bracket, expand the bracket, sorry.

So again, x squared plus 5x plus six.

That's now in the form axe squared plus bx plus c.

Okay, this one.

Let's get this in the form axe squared plus bx plus c.

So the first thing I'm going to do is, I'm going to expand that bracket.

Okay, now I'm going to rearrange it.

Well, I've got my squared on that side.

So I'll tear these off this side.

And it helps, I always think, to write the thing we're taking away, below, so like if we're taking away x's, to write it below the x's.

So rather than writing it there, because then I might have took them away, kind of like we line things when we're doing, in fact, exactly like we line things up when we're doing subtraction in a column.

So it gets zero on that side equals 6x squared, plus 10x minus one.

Okay, and now I'm just going to do a bit of a switcheroo here.

Okay, so remember if something is equal to something, we can swap those somethings around and it does not affect our equation.

So that's in the form x squared plus bx plus c.

And finally, how do I get rid of this x on the denominator? I multiply by this x, okay? We can multiply by the x to get rid of x on the denominator.

Okay, I'm multiplying by that.

So that's three divided by x, times x.

The divide by x and the times x cancel out or simplify to one so that we just get a three.

So then we can rearrange that by taking away three from both sides.

There was a quite a lot there.

All right, and I went through that quite quick.

So I would like you to use, I'd like you to use what I've done and try and answer these.

Okay.

So pause the video and have a go.

Pause in three, two, one.

Okay.

So welcome back.

Now, let me go through my answers.

So I'd expand this and you should have got x squared plus 9x plus eight equals zero.

On this one you should have got, and then take these from that side.

So we get zero equals 20x squared, and I didn't write it over here, silly me.

Plus 5x minus four, and then do a bit of a switcheroo and there we are, okay.

And now I multiply both sides by 3x.

So I get 2x times 3x is 6x squared, plus four times 3x equals five.

Subtract five, subtract five.

And there we go.

So really well done if you got those correct.

Okay, so now we are going to write the following quadratics in the form x squared plus bx plus c.

And this is the independent task.

So I would like you to try questions 1, 2, 3.

So pause the video to complete your task and resume once you're finished.

Okay.

So here are my answers.

You may need to pause the video to mark your work.

Okay.

So now it is time for the explore task.

Okay.

So axe squared plus bx plus c equals zero.

That's our quadratic equation.

Now I've told you that a times b times c equals 24.

If a, b and c are positive integers, what are the possible values of a, b and c? Have a think about that.

Write three different quadratic equations where a plus b plus c equals 12.

So when I say, "what the possible value of a plus b plus c," I mean, adding those constants together.

So adding the a and the b and the c.

So if a was five and b was five and c was seven, we do five plus five plus seven.

And for the final part, write three different quadratic equations where a plus b plus c equals 12.

And they multiply together to get 24.

Okay.

A bit tricky this, but have a go.

So pause the video to complete your task.

Resume once you've finished.

Okay, so welcome back.

Here are my possible sums. So if they add to 24, they must be the factors of 24.

So we have a times b times c equals 24.

So we could have had one, one and 24, which would be, give me 26.

We could have had, obviously you could, I'm not going to list them all now, but you could have had two times three times four, that equals 24.

So two plus three plus four, they add together to get nine.

So we've got our nine and 26, and you could find the others as well.

So if you haven't worked out how to get the others and you couldn't do that, maybe it's worth you having to go at it now and write three different quadratic equations where a plus b plus c equals 12.

And a times b times c equals zero.

So that's where they equal to that.

So what was the values of a or what were the three numbers that times together to get 24, and add together to get 12? Think about them.

Okay.

Once we've got them, how can we write some different equations with them? So if you haven't done this yet, pause the video and complete it now.

Otherwise, a plus b plus c, I think that was, I can't remember now.

I can't remember off the top of my head now.

I think it was one, three and eight.

Yes, it was one, three and eight.

So here, my coefficient is one, three and eight.

Now here, I've got a bit creative.

So here, I've done three, eight, and that's not negative one because when I add it to the other side, that will be a one.

And here I'd have to expand that bracket and then subtract that from the other side and when I do that, they would give me my coefficients of one, three and eight.

And they don't have to be in that order.

They could be three, eight and one; three, one and eight, et cetera.

Okay, so hopefully you managed to find a few of them and hopefully you enjoyed that.

So that is all for this lesson.

Thank you very much for all your hard work.

And I look forward to seeing you next time.

Thank you.