Lesson video

Hello, I'm Mr. Coward, and welcome to Forming Quadratic Equations, Part II.

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

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

And if you can, please try and 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.

Find the area of these shapes in terms of X.

So you may need to use your knowledge of expanding double brackets.

Okay, so pause the video and have a go.

Pausing three, two, one.

Okay.

Welcome back.

So, I need to work out the area of this rectangle, so I use base times height.

Now, it doesn't actually matter which order you do it in, or which one you consider your base because multiplication is commutative.

I'm going to do it this way though.

I don't know entirely of how, how you've been taught how to expand brackets, but I'm just going to do it my way.

Okay, so I'm going to do my 3X times by my X, my 3X times by my 1, my 5 times by my X, and my 5 times by my 1.

And then I collect like terms. Which in this case is just the X's.

Okay, so that is the area of the rectangle.

And this is a quadratic expression.

Okay.

Why is it an expression? Well, because it's not equal to anything.

And why is it quadratic? Well, the highest power is two.

Okay, so the next one.

Well, what's my formula for an area of a triangle? My formula is half base times height.

And so people think of it as base times height divided by two.

Well, I prefer it this way because you can use the commutativity of multiplication and the associativity of multiplication here, and I'm just going to do whatever's easiest.

So what I'm going to do first is I'm going to do half times my base, because I could do times the half at the end, but I think that would be slightly more complicated.

So when I half this length here, half of 2X, 1X, half of four, two.

Okay, now let's expand that bracket, that times that, that times that, that times that, not forgetting about the sign, and that times that.

Okay and collect like terms, which in this case is just the X's.

Okay.

So, that is my area of the triangle.

It's an expression because it's not equal to anything.

And the highest power is two.

So it's a quadratic.

Okay, well this one, what's this called? This length here.

It's called a diameter.

Now, it's easier to find the area of the circle using the radius.

So I'm going to find my radius.

So if that's the centre, that point there, what's that length? It's half of my diameter, half of 6X, which is 3X.

So, now I can use the formula pi r squared to find my area of this.

So pi times the radius squared and the radius is 3X.

So now we've got pi times 9X squared, which we prefer to write like this.

We prefer to write our numbers first, then our constant symbols like pi, and then our variables at the end.

And if you wrote pi 9x, you are not wrong.

But it's just preference.

We just prefer to write it like that in a standard way.

Okay, so some of these, these were pretty tricky, you know, so really well done if you managed to get a couple of them, but we're going to kind of be building upon these skills in today's lesson.

So you'll get plenty of opportunity to practise if you struggled.

Form quadratic equations for the area of these shapes.

Ah, so this is very similar to what we were doing.

But we don't want an expression anymore.

We want an equation and we can do that.

We can form an equation because we know what the area is.

So, let's work this out.

So to find my area, an expression for my area, I would do that.

And that would be equal to 23.

So I worked that out, which is the same one from before.

So we get our 3X, collect like terms. So now that is our area.

So what did we do? We multiplied the two lengths, the base and the height.

And we set that equal to our area.

'Cause that is just how we find our area, base times height equals area.

Then we expanded our brackets and we collected like terms. And that is still obviously equal to the same thing, because we've just, we've just written that in a different way.

We haven't changed the value of that at all.

So here we have a quadratic expression for our area.

Okay, so something that's quite important here is that we got a quadratic because we times two lengths together, which both had the same unknown in.

Okay? So what that means is this length and this length, they both had an X in.

So at some point when we were multiplying them together, we did X times X and we needed that.

We needed to do the X times X to give us a quadratic.

If one of the lengths didn't have an X in, then we would not have got a quadratic.

Okay, so what I would like you to do is I would like you to have a go at this one.

So pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

So, here are my answers.

So, times the base and the height and that'll help me get the area or that will find the area for me.

So that times that, so I've got a 12X squared, that one times that one.

That's a lovely two, isn't it? And 6X, 3 times 2X, 6 again, and plus 3 equals 35.

I collect like terms. So I get 12X squared plus 12X plus 3 equals 35.

That is an awesome three.

Okay.

So really well done if you've got that correct.

And if you didn't, hopefully you've spotted your mistake.

Okay so now we're going to find, we're going to write an equation for the area of this triangle.

So base times height, divided by two, or half times base times height.

And like before, in fact I'm going to do it different from before.

I'm going to half this one, okay? Because it doesn't matter.

So half of that one would be 2X plus 3.

So I'm going to do, okay, so that will give me 4X squared minus 8X, plus 6X, which is, yeah, just checking there.

Okay.

Now that is equal to our area, that is equal to 23.

So now just that addition of halving the area, that expression has become an equation, okay? So that, knowing what that area is, has meant that that expression has become an equation because it's equal to something and it's a quadratic equation.

Okay, and it's quadratic because we timed at some point, we times two X's together and the 2X and the 2X, which gave us 4X squared.

So because we multiply two X's together, we get a squared and that's what made it become quadratic.

Okay, so your turn.

Have a go at this one.

So pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Now I will work through this.

So I'm going to half, I'll half this one.

So I've got X plus 2, and that is equal to 23.

So now I expand that, 6X squared.

Okay, hopefully you expanded this correctly.

And then collect like terms. Okay.

Did you get that? If you didn't, did you find your mistake? Really well done if you got it.

And if you didn't, hopefully you found your mistake.

Okay.

Form a quadratic equation for the area of these shapes.

So this time we have circles.

So we need to use our formula for a circle, area of a circle, pi r squared.

So our area, which is 23, is equal to pi times by the radius squared.

Well, that's not the radius, is it? That's the diameter.

Hmm.

So what's our radius? Our radius would be half of that.

So 2X plus 1.

So we have to square that.

Okay, so we'd expand those brackets.

Okay, 'cause remember that squared just means that thing times by itself.

So that's why I like to write it when I'm squaring out those two brackets.

So we've got pi times, and I'm going to put brackets around this, because it's pi times everything.

Okay? It's not just pi times just the first term, it's pi times everything.

So I need to write a bracket around it so that it makes sure that I times everything by pi.

So here I get 4X squared, plus 2X, plus 2X, plus Y.

So my final answer is going to be pi, I don't need to write my multiplication sign because bracket means multiply.

Okay, so pi, 4X squared, plus 4X, plus Y.

Okay, have a go at the your turn one.

So, pause the video and have a go.

Pause in three, two, one.

Okay.

So hopefully you've had a go.

Hopefully you've found that the radius is X plus 2 squared.

I don't need to write my time sign, 'cause bracket means multiply.

So we get X squared plus 4X.

Okay.

I just skipped a step there.

So it would have got, we would have had X plus 2 times X plus 2.

So, and you expand that and you get X squared plus 4X equals 4.

Okay, so I've jumped a step just because I'm conscious of space.

Okay, so now that is my answer.

But the reason why I was conscious of space is because I want to show you that you could also write your answer like this.

You could also expand that bracket and write it with the pi on the inside.

Now you don't have to do that.

It's just another way of writing it.

So it depends what you're looking for.

So both of these are correct.

And which one do you prefer, really? Which one do you think looks nicer? Which one do you think is more simple? In this situation, I'm not sure.

In some situations I might have a preference, but it's important that you recognise that that and that are the same.

Okay.

Final one.

And this is probably the trickiest one for quadratic equations, form quadratic equations for the length of the hypotenuse squared.

What do we have to use here? We have to use Pythagoras.

I've got a right angle triangle, and we're trying to find what that length is in terms of that and that.

So what I'm going to do is I'm going to square this length.

I'm going to square this length.

And because I'm squaring two lengths, I know my equation is going to be a quadratic.

So that squared plus that squared will be equal to 13 squared.

That is Pythagoras's theorem.

So now I need to work out what that squared is.

13 squared is 169.

So that times that, so X times X, X squared, X times 3, X times 3, plus nine.

Yeah, we do the same thing and we get X squared plus 4X plus, sorry.

Plus 4X plus 4X plus 16 equals 169.

Okay, now we collect our like terms. So we have 2X squared plus 6X plus 8X, 14X, 9 plus 16, 25 equals 169.

Okay, so here we have our quadratic equation.

Alright.

So that was quite a few steps there.

So let's talk through it.

So we use Pythagoras to say that the square of the two shorter sides is equal to the square of the longest side.

A squared plus B squared equals C squared.

So then we actually squared them.

And I just wrote, I like to write them out, that out, X plus 3 squared is X plus 3 times X plus 3.

And I have to write out X plus, X plus 4 squared is X plus 4 times X plus 4.

And 13 squared is 169.

So then I square that.

I work that out.

So I have expanded those brackets.

And then I collect like terms. So we've got X squared plus X squared, 2X squared, 3X plus 3X, 6X plus 4X, 10X plus 4X, 14X, and then 9 plus 16, which is equal to 25.

Okay, so your turn.

So pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Now I am going to speed through this.

So I square this side and that side and that is equal to 7 squared.

So that squared is this.

I've got a really neat trick for, actually squaring brackets.

I wonder if you know it, too.

If you don't, don't worry.

'Cause it is just a trick and it's, you know, it's easy to forget tricks, but maybe you can work it out.

Okay, so here we've squared both brackets.

Now I collect like terms. So I get 2X squared plus 14X plus 29 equals 49.

Okay.

There we are.

So 2X squared plus 14X plus 29 equals 49.

Really well done if you got that correct.

Okay.

Just want you to pause the video and have a think about this.

Okay, so which one would form a quadratic equation and why? Well, this one would.

Why? Because on this one, to work out the area we'd do that.

And we times two X's together.

And so, because we are multiplying two X's together, we would get a squared.

Where on this one, we'd be doing this calculation.

And we'd be multiplying 7 by an X, but 7, oh sorry, 77 there.

But that, that does not give me a quadratic because it's a known value times by my variable.

So it doesn't make that squared.

Whereas this one, we're times-ing an X by an X.

So we get an X squared.

And add 77x, equals 42.

Okay so this one is a quadratic.

This one is not a quadratic.

Okay, so now it is time for the independent task.

I would like you to pause the video to complete your task and resume once you're finished.

Okay, welcome back.

Here are my answers.

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

Okay, so now it's time for the explore task.

So the surface area of this is equal to 200 centimetres squared.

Do you remember what the surface area is? Well, it's the area of all the surface.

So the area of the outside.

Okay.

All of its faces.

The volume equals 420 centimetres squared.

Do you remember how to work out the volume of the cuboid? Hmm.

Okay, how do we do that? Form two equations.

What do you notice about your equations? Okay, so I'd like to pause the video and have a go at this task and resume once you've finished.

Okay, and here are my answers.

Okay, so the surface area.

So the area of all the surface.

That is one, two, three, four, five, six faces.

Now I'll start with this face.

It's a nice, easy square.

So X times X is X squared.

And that is the same as that face there.

So I have X squared plus X squared, which is 2X squared.

Now this length is X, X times X plus 11X squared, plus 11X.

That length is X plus 11.

That is X.

And that side is the same as the bottom.

And that side is the same as that.

So because this is a square here, I actually have four sides that have the same area.

So I have four times X squared plus 11X.

So expand that into 4X squared plus 44X.

And that gives me my final answer of this.

Okay, the volume.

Well, I have to do, I'm going to do the area of this face.

I could have done any faces 'cause it's, it's a cuboid, but I'm going to do this face, so that's X squared times by that length there, and so X squared times by X plus 11 gives me this.

So let me just write that down for you.

X squared for that face times by the other length, X plus 11.

So I get X cubed plus 11X squared equals 420.

Now, what is the difference between our two equations here? Well, this one, this one gave me a cubic, and this one gave me a quadratic.

Why? Well, this one, I just multiplied two lengths together.

Okay.

So two, 'cause I multiplied two lengths to get the area.

And then I summed those areas.

So I multiplied the variable by itself, once to get a squared.

Whereas here I multiplied the variable by itself twice, which gave me a cubed.

So X times X, times another X here.

Now, the volume doesn't always have to be a cubic.

So for instance, if that length at the bottom was not X plus 11, but if it was just 11, well then our volume would be 11X squared.

So we would sometimes get a cubic.

It just depends, and the thing that I want you to look out for is how many unknowns are we multiplying together.

Okay, how many of that same variable are we're multiplying together? And that will determine whether it's, it's linear or it's cubic, or it's quadratic or even a quartic, okay? Which is a power of four.

All right, so that is all for this lesson.

I hope you enjoyed it and I hope you find it, found it interesting.

Thank you very much for all your hard work, and I will see you next time.

Thank you.