Lesson video

Hi, I'm Mrs. Wheelhouse and welcome to today's lesson on factorising a quadratic expression.

This lesson is part of our unit on algebraic manipulation.

Now, algebra can be extremely useful, so let's find out how we're going to be using it today.

By the end of today's lesson, you'll be able to factorise quadratics of the form X squared plus BX plus C.

Now, these are some keywords we're going to be using in our lesson today, and I suggest you pause the video and have a read through these now so they're familiar to you.

Our lesson today has three parts, and we're going to begin by factorising using algebra tiles.

To factorise an expression is to write it as a product of two or more expressions.

For example, three lots of X plus 4 is a factorised expression as it's the product of 3 and X plus 4.

Also, X plus 4 multiplied by X plus 5 is a factorised expression as it is the product of X plus 4 and X plus 5.

It can be useful to write an expression as a product of expressions rather than a sum.

A quadratic expression is an expression whereby the highest exponent of the variable is two.

And here are some examples, and non-examples.

Now, some quadratic expressions can be factorised, and we're going to focus on quadratic expressions in the form X squared plus BX plus C, such as, for example, this one here, X squared plus X subtract 2.

What expression do these algebra tiles represent? That's right, it's X squared plus 5X plus 4.

Now, they've been positioned so they form a rectangle.

What are the dimensions of the rectangle? So in other words, what's its length and width? That's right.

We have X plus 4 and X plus 1.

So this area can also be expressed as X plus 1 multiplied by X plus 4.

And this is the factorised form of X squared plus 5X plus 4.

Let's do a quick check.

Please write the expression represented by these algebra tiles in expanded form and factorised form.

Pause and do this now.

Welcome back.

Let's see how you got on.

Well, expanded form, we have X squared plus 6X plus 8, and in factorised form, we have X plus 4 and X plus 2.

Now, of course, you could write these the other way around, and it's absolutely fine if you did.

Sofia is considering some algebra tiles.

They represent the expression X squared plus 4X plus 4.

How could Sofia arrange these so they form a rectangle? Now, if you have some algebra tiles, feel free to have a go right now, or you could try drawing them.

What arrangements did you come up with? Well, this is one of the ones Sofia came up with.

And don't forget, you could still have arranged it correctly, just it not look exactly like Sofia's 'cause you can have different orientations.

But what you should reach is a rectangle that has the same dimensions that Sofia does, which is X plus 2 and X plus 2.

Now, you could, of course, write this as X plus 2, all squared.

Now, Jun wants to factorise X squared plus 7X plus 6.

So we've got our algebra tiles, and remember, we need to arrange them into a rectangle.

Oh, he's got this far, but now the X terms don't fit.

Think about it, he's only got space for three X terms and he's actually got five.

So what can he change to make this work? Well, he's put his six 1 tiles in a three by two array, but he could arrange them into a six by one array.

Let's see what that looks like.

Ah, if he does this, he now has space for the remaining six X tiles.

He's got one more this time, because remember, we don't have a three by two array, we now have a six by one array.

so the one X tile that was sitting on the top has to move.

And there we go.

Now, we have a rectangle.

So the factorised form of X squared plus 7X plus 6 is X plus 6 and X plus 1.

Which of these show a correct representation of the factorised form of X squared plus 7X plus 10? Pause and make your choice now.

Welcome back.

Which ones did you go for? Ah, it's C, and in fact, C is the only one.

If you look very carefully, you'll notice that some of these diagrams have too many X tiles, and some of them are not arranged in the correct way.

We do not know the length of X, so when building rectangles, we should never place ones tiles along the edge of an X squared or along the long edge of an X tile.

So let's use this diagram to factorise X squared plus 7X plus 10.

And then for the bottom diagram, we know it doesn't represent a factorised form of X squared plus 7X plus 10, but what does it represent? Can you write it in both expanded and factorised form, please? Pause and do this now.

Let's begin with writing the factorised form of X squared plus 7X plus 10 using the diagram.

Well, we can see the dimensions are X plus 5 and X plus 2.

What expression could the diagram at the bottom represent? Well, we have X squared, and there are 11 X tiles, so plus 11X, and then 10 one tiles.

So we have X squared plus 11X plus 10.

In factorised form, that's X plus 10 and X plus 1.

It's now time for your first task.

Please use the algebra tiles to write each of these in factorised form.

Pause and do this now.

Welcome back.

Question two, Jun wants to factorise X squared plus 8X plus 12.

Now, Jun says, "I'm going to have to try lots of different arrays with my ones tiles this time." What do you think he means by this? And then part B, how will you know which array of 12 to use? And then in C, please show the correct factorisation of X squared plus 8X plus 12.

Pause and do this now.

Time to go through your answers.

On the screen, you can see the correct rearrangement of the algebra tiles for each of the questions.

For 1A, you should reach the factorised form of X plus 3 multiplied by X plus 3.

In B, we have X plus 1 multiplied by X plus 3.

In C, X plus 2 multiplied by X plus 3.

And in D, X plus 1 multiplied by X plus 5.

Question two, what did Jun mean by lots of different arrays for his ones tiles? Well, 12 has three different factor pairs.

He could form a one by 12, a two by six, or a three by four array.

So he's got a lot to test out here.

So how will he know which array to use? Well, he needs eight X tiles, so it will be the one where the dimensions add to eight.

So here, we have the correct factorisation of X squared plus 8X plus 12.

You can see how I've arranged the algebra tiles so that I can see that I have X plus 2 multiplied by X plus 6.

It's now time for the second part of our lesson, and that's on exploring factorising with negative terms. We can place algebra tiles on a set of axes to make sense of negative lengths.

Remember, positive values appear here on the axes, and negative values are here on the axes.

So if this was a multiplication grid, where would the products be positive? They'll be positive here because positive times positive is positive and negative times negative is positive.

And the products would be negative in the other two quadrants.

When all terms are positive, the area model can be drawn in the first quadrant, as we've seen previously.

So this diagram is not placed solely in the first quadrant.

What expression is represented by these algebra tiles? And why have the terms been positioned this way? Well, let's consider it.

First of all, I have X squared.

There are five lots of negative X tiles, so that's negative 5X, and then six ones.

In our factorised form, we have X takeaway 2 and we have X takeaway 3.

So we multiply those two together to get the area.

Here is where we can see the X takeaway 2.

And here's where I can see the X takeaway 3.

The constant 6 is positive, so placing it in that quadrant can help us make the rectangle.

The positive algebra tiles are in the positive quadrants and the negative algebra tiles are in the negative quadrants.

Which of these is a correct positioning for the expression X squared minus X minus 6 on a set of axes? Pause and make your choice now.

Welcome back.

You should have picked this one.

Now, this is the only diagram where the positive algebra tiles are in a positive quadrant and the negative algebra tiles are in a negative quadrant.

It is also the only one that correctly shows X squared minus X minus 6.

So let's use the algebra tiles now to write X squared minus X minus 6 in factorised form.

Pause and do this now.

Welcome back.

You should have seen that we have X plus 2 and we have X minus three, so they should be what are in our brackets.

You could, of course, have also drawn the diagram this way.

These two diagrams are equivalent.

They're just in a different orientation.

Now remember, the X minus 3 and the X plus 2, those could have been written either way around.

What expression is represented by these algebra tiles? Please write your answer in both expanded and factorised form.

Pause and do this now.

Welcome back.

Well, we have X squared.

We then have two zero pairs of X, so we're left with just one positive X.

And then I have six negative ones tiles.

So I have X squared plus X subtract 6.

I can see that dimensions for this rectangle are X plus 3 and X minus 2.

So in factorised form, I've written X minus 2 multiplied by X plus 3.

What you can see here are some different arrangements of algebra tiles along with the expanded form they represent and the factorised form they represent.

Take a moment to review this and see what is the same and what is different about each representation.

Alex wants to factorise X squared plus 2X subtract 8.

If he uses a pair of axes, where can he place the constant term of negative 8? That's right, there are two places, because there are two negative quadrants.

What options does he have for an array with the constant term? That's right, he can make a two by four, a one by eight, a four by two, and an eight by one.

So he's got lots of arrangements.

Now, this is where Alex chooses to place some of his tiles.

And he points out that no matter how he arrange these, he doesn't have enough X tiles, and he's right.

What could Alex do though? That's right, he could fill these gaps with some zero sum pairs.

So let's place these tiles.

There we go.

Now, we have a rectangle and we still represent the same expression.

So what is the factorised form of X squared plus 2X subtract 8? That's right, it's X plus 4 multiplied by X minus 2.

Let's do a quick check now.

Please complete these models to show the factorisation of X squared minus 10X plus 9, and X squared minus 4X minus 5.

Pause and do this now.

Welcome back.

Let's consider A first.

Well, in order to do the array for the 9, I know that it had to be a one by nine.

And that's because if you look back, four tiles were already placed, so I had to place the other five there because there wasn't any other way to arrange the 9's ones tiles.

I couldn't do a three by three array because four were already placed.

This then told me I had to place more negative X tiles there.

And then when I looked, I realised I had all the negative X tiles I needed.

So therefore, the factorised form must be X subtract 9 and X subtract 1.

For part B, if you looked before, we already had three negative ones tiles placed, so I knew I had to place these in a five by one array, which is what I can see here.

I then had to fill in the gaps.

And when I did that, I'd ended up with five negative X tiles and I had a space where I needed to put a positive X tile.

Well, when I get rid of the zero sum pair, that still leaves me with four negative X tiles.

So the factorised form of X squared take away 4X takeaway 5 is X subtract 5 multiplied by X plus 1.

It's now time for your second task.

Use zero pairs to complete these diagrams and then write the factorised form for each quadratic.

Pause and do this now.

And question two, use the algebra tiles and any necessary zero pairs to factorise these expressions.

Pause and do this now.

Here are the answers to question one.

What you can see on the screen are the completed diagrams, and then underneath, the factorised form.

So for 1A, you should have X plus 3 multiplied by X minus 1.

1B, you should have X plus 2 multiplied by X minus 5.

And 1C, X plus 5 multiplied by X minus 4.

Remember, you may have written these around the other way and it's absolutely fine if you have.

For 2A, you should have X minus 2 multiplied by X minus 4.

For B, X minus 4 multiplied by X minus 1.

And for C, X plus 8 multiplied by X minus 1.

It's now time for the final part of our lesson today, and that's on using area models to factorise quadratics.

Now, it's not always efficient to use algebra tiles to factorise quadratics.

So instead, we can turn our algebra tiles into area models.

And these do not have to be to scale What expression is represented by this area model? That's right, we have X squared, we have negative 6X, negative 3X, and then we have 18.

We can write this in factorised form as X subtract 3 multiplied by X subtract 6, or in expanded form as X squared minus 9X plus 18.

Here is a factorised expression.

How could we write this in expanded form? Well, we know we can use our area model.

We have X lots of X minus 12, which is X squared minus 12X, and we have five lots of X minus 12, which is 5X takeaway 60.

We would then gather any like terms, which are the negative 12X and the 5X, to give us X squared minus 7X minus 60.

Now, we can compare the terms in our simplified quadratic expression with the area model.

We can see where the X squared term is.

We got that from doing X multiplied by X.

We can see where the negative 60 is.

We got that by doing 5 times negative 12.

And we can see where the negative 7X came from.

That was from summing 5X and negative 12X.

We can use the area model to factorise quadratics.

Let's consider X squared plus 6X minus 16.

We know what our X squared term and our constant term are, and we can fill them into our area model right now.

We can now label the dimensions of the top left rectangle.

We know in order to reach a product of X squared, we must have multiplied X by X, because it's a square.

Now, the binomials must be in the form X plus something and X plus something else, and those somethings could be positive or negative values.

So what could the other terms in our binomials be? Well, we know there need to be two values which multiply to give negative 16.

So let's consider all the factor pairs.

Now, I've got 1 times negative 16, 2 times negative 8, and 4 times negative 4, but I need to consider what if the other factor was negative? In other words, 16 times negative 1, and 8 times negative 2.

There's no point writing out negative 4 times 4.

It's the same as 4 times negative 4.

So we know that one of these pairs is going to be correct.

It's like when we were considering all the different possible arrays of the ones tiles when we were working with algebra tiles.

We're doing the same thing here.

Now, we know that what goes here in the two remaining areas must be like terms that sum to 6X.

So which of our options are going to work? Remember, we want to sum to make 6.

Which factor pairs will work here? That's right, it'll be 8 and negative 2.

8 multiplied by negative 2 gives us negative 16, and 8 plus negative 2 gives us 6.

So we're going to put those on, and then we'll just check that this does calculate the way we think it will.

Well, negative 2 times X is negative 2X, and X times 8 is 8X.

So we know that the factorised form must be X minus 2 multiplied by X plus 8.

And we can see our four partial products in our area model and we know that if we gather the like terms, we'll get back to our original expanded form.

Let's try factorising X squared minus 12X plus 35.

So we can put in the X squared term and the constant term.

We then have to say to ourself, "We know what the form of the binomials must be, so we know the constants have to be positive as they must multiply to make 35," says Sofia.

Do you think she's right? Well, she's correct they have to multiply to make 35, and it is a positive 35, but we have to consider that two negative values also have a positive product.

So there are different options here.

Now, if you've been watching carefully, you may have already noticed that I don't want to consider two of these automatically.

Now, why are two of these pairs no good to me? Well, they have to sum to make negative 12.

If I've got two positive values, that's not going to sum to make a negative.

So I know that I'm either going to be using negative 1 and negative 35 or negative 5 and negative 7.

It's going to have to be the negative 5 and negative 7.

They're the only two that sum to negative 12.

So therefore, the factorised form must be X subtract 5 multiplied by X subtract 7.

And again, we can check this by considering the four partial products in the area model.

They do indeed simplify to X squared subtract 12X plus 35.

Let's do this one together.

So I'm going to put in the X squared and the negative 21.

I'm looking for the factor pairs for 21.

Remember, one of them will need to be negative because I have a negative product.

So here are all my different factor pairs.

Now, the factor pair I pick, the two values must sum to give negative 4.

Can you spot which ones I'm going to choose? That's right, it's 3 and negative 7.

Now, I'll fill in the partial products and I'll write the factorised form of X plus 3 and X minus 7.

And I just check with the area model, 3X subtract 7X is negative 4X? Well, it is indeed.

And now, you can even do this by expanding to make sure.

There we go.

I get back to the expanded form.

So this must be the correct factorisation.

It's now your turn.

Please factorise X squared minus 3X minus 10.

Pause and do this now.

Welcome back.

How did you get on? Well, you should have put in the X squared to the area model, and put in the negative 10, considered the factor pairs for negative 10, and worked out, for each of these pairs, do the two values sum to make negative 3? The pair you should have chosen, therefore, are 2 and negative 5.

So the factorised form is X plus 2 multiplied by X minus 5.

And again, you could then have expanded this to check you got back to X squared minus 3X minus 10.

For this one, I'm going to do X squared minus 16X plus 15 and I'm going to factorise this.

So I fill in the X squared and the positive 15.

Now, I know from my factor pairs that both are going to have to be negative because they sum to negative 16.

So I'm going to choose negative 1 and negative 15.

So my factorised form will be X subtract 1 multiplied by X subtract 15.

And again, I can check this by expanding.

I do indeed get back to X squared minus 16X plus 15.

It's now your turn.

Please factorise X squared minus 11X plus 18.

Pause and do this now.

Welcome back.

Did you use your area model? You should have filled in the X squared and the 18.

You're looking at factor pairs of 18 and both will have to be negative because we must sum to make negative 11X.

So which pair are you going to choose? Well, it should be the negative 2 and the negative 9.

So the factorised form will be X subtract 2 multiplied by X subtract 9.

And you can again check by expanding, and you do indeed get back to X squared minus 11X plus 18.

Well done if you got this right.

It's now time for your final task.

Please fill in the missing terms in the area models and then factorise each expression.

Pause and do this now.

Welcome back.

Question two, use the area models to factorise each expression.

You'll notice I've given you less hints this time.

Pause and do this now.

Welcome back.

Question three, factorise each expression.

You can draw an area model to help you if you wish.

Pause and do this now.

Welcome back.

Let's go through our answers.

For question one, you had to complete the area models and factorise each expression.

So you can see on the screen what the answers are.

I'm going to suggest you pause now so that you can check your working against what I have on the screen.

Because there's a lot here to check, and reading them out might be a bit confusing.

Pause and do this now.

Question two.

I'm going to give you the same advice.

I suggest you pause the video because there's a lot of working on the screen here, so that you can check what you have with what I have.

Pause and do this now.

Now, question three, you had to factorise each expression.

For 3A, you should have reached the factorised form of X plus 3 multiplied by X plus 2.

In B, we have X subtract 3 multiplied by X subtract 2.

3C, X subtract 6 multiplied by X plus 1.

D, X subtract 1 multiplied by X plus 6.

E, X subtract 12 multiplied by X plus 2.

F, X subtract 6 multiplied by X minus 4.

In G, X plus 6 multiplied by X plus 4.

And H, X subtract 2 multiplied by X plus 12.

Well done if you've got all these right.

Let's sum up what we've learned today.

To factorise an expression is to write it as a product of two or more expressions.

Arranging algebra tiles into a rectangle or using an area model can help us see the factors.

Working backwards from an area model using the terms in our quadratic allows us to find the factors of the quadratic.

Knowledge of directed numbers can help us select the correct factors.

We can check our factors by finding the product and checking it against the original quadratic.

Well done.

You've worked really well today.

And I look forward to seeing you for more lessons on algebraic manipulation.