Lesson video

Hi, my name's Mr Clasper, and today we're going to learn how to factorise a quadratic expression.

Let's begin by expanding this expression by using a multiplication grid.

X multiplied by X would give us X squared.

5 multiplied by 5 would give us 5X.

3 multiplied by X would give us 3X.

And 5 multiplied by 3 would give us 15.

This means that our expression is equivalent to X squared plus 5X plus 3X plus 15.

We then need to collect our like terms, which means that our expression will be X squared plus 8X plus 15.

You may want to make a note of the fact that 5 multiplied by 3 gave us our constant of 15, and 5 plus 3 gave us our coefficient of X, which was 8.

Let's factorise the expression X squared, plus 5X plus 6.

We know from our previous example that we need two numbers which will multiply to give us 6, and add together to give us 5.

Let's have a look at the factor pairs of 6.

We could use 1 and 6, or we could use 2 and 3.

Let's try 1 and 6.

Using the multiplication grid, this will give us X squared, plus 6X, plus X, plus 6.

This will not work, as when we add our like terms together, we would get 7X, and we want 5X.

So 1 and 6 will not work.

Lets' try 2 and 3 instead.

When we try 2 and 3, using the multiplication grid, we would get X squared, plus 2X, plus 3X, plus 6.

And as we can see, 2X plus 3X would give us 5X, which is what we wanted.

So when we factorise X squared plus 5X plus 6, we get X plus 2, multiplied by X plus 3.

Here are some questions for you to try.

Pause the video to complete your task, and resume once you're finished.

And here are you answers.

So take note that for part A, we got 3 and 5 in our bracket, and the product of 3 and 5 is 15, and the sum of 3 and 5 is 8.

And also for part B, we got four and 3 in our brackets, and the product of 4 and 3 is 12, and the sum of 4 and 3 is 7.

We're going to use that shortly.

Let's have a recap.

Earlier we factorised X squared, plus 5X, plus 6.

Looking at this further, to achieve this, we needed two numbers which had a product of 6, which was our constant, and the sum of 5, which was our coefficient of X.

Let's factorise the expression X squared, plus 7X, plus 12 using a similar method.

To factorise this successfully we need two numbers which have a product of 12, and a sum of 7.

Let's have a look at the factor pairs for the number 12.

We could use 1 and 12, however these do not have a sum of 7.

So we cannot use 1 and 12.

We could use 2 and 6, however they do not have a sum of 7X, therefore we cannot use these two.

3 and 4 have a product of 12, and they also have a sum of 7.

Therefore these are the two numbers which I can use to fully factorise my expression.

Here's a trickier example.

Let's factorise X squared, plus 8X, minus 20.

For this example, we're looking for 2 numbers that will multiply to give us negative 20, and add together to give us 8.

Let's begin by looking at the factor pairs for negative 20.

We could have positive 1 and negative 20, or negative 1 and positive 20.

We could use positive 2 and negative 10, or negative 2 and positive 10.

Or we could use positive 4 and negative 5, or negative 4 and positive 5.

What we need, are a pair of these numbers which have a sum of 8.

So 1 and negative 20 do not have a sum of 8.

Negative 1 and 20 do not have a sum of 8.

2 and negative 10 do not have a sum of 8, these have a sum of negative 8, so we cannot use these.

However, negative 2 and 10 do have a sum of 8, so we're going to use these two numbers.

That means that X squared, plus 8X, minus 20 will be equal to X minus 2, multiplied by X plus 10.

Let's look at one final example, let's factorise X squared, minus 16X, plus 15.

For this example we're looking for 2 numbers with a product of 15, and a sum of negative 16.

Let's find the factor pairs of 15.

We have 1 and 15 and 3 and 5.

1 and 15 have a sum of positive 16, not negative 16, so we cannot use these two.

3 and 5 have a sum of positive 8, so that means we can't use these two either.

We have to look a little bit further for this one.

Another pair of factors we could use would be negative 1 and negative 15, as when we multiply these together we get positive 15, and we could also use negative 3 and negative 5, as these also have a product of positive 15, which is what we wanted.

The sum of negative 1 and negative 15 is negative 16, so this is the pair of numbers which we're going to use to factorise.

Negative 3 and negative 5 would have a sum of negative 8, which is not what we want.

Here's a question for you to try.

Choosing A, B, C or D, can you identify the correct answer? Pause the video to complete this task, and resume once you're finished.

Did you find the correct answer? The correct answer was D.

So two numbers which have a product of negative 42 and a sum of negative 1, would be negative 7 and positive 6.

Well done if you got that one right.

Here are so questions for you to try.

Pause the video to complete your task, and resume once you're finished.

And here are your answers.

Looking carefully at D we can see that we need a product of negative 10 and a sum of 3, so we needed to think carefully about those two numbers.

And likewise for F, we have a positive product, so we needed a product of positive 12, but a sum of negative 7, which meant that we had 2 negative values inside our bracket.

Here is you final question.

Pause the video to complete your task, and resume once you're finished.

And here are your answers.

If we look carefully at the top left, the top left when we expand it would give us X squared, plus 7X minus 18.

And top right will give us X squared, minus 11X, plus 18, which isn't equivalent to the expression given, so it's the bottom two which are correct.

And that's it for the lesson.

Thank you very much for your time.