Lesson video

Hi, my name is Mr. Clasper.

And today we're going to learn how to factorise a quadratic with a difference of two squares, which also has a leading coefficient which is greater than one.

Let's begin with an example of factorising when there's a difference of two squares.

The expression given could also be written as x squared plus zero x minus 16.

So this means we need two numbers with a product of negative 16, and the sum of zero.

So looking at our factor pairs for negative 16, we could choose one and negative 16, negative one and 16, or four and negative four.

If we choose one and negative 16 in this order, we would not get a sum of zero.

So we need to think more carefully about this.

If we chose four and negative four, this will give us a sum of zero as four x subtract four x would give us our zero.

Let's try this example.

You should notice that the leading coefficient is no longer one.

So in our case, we have a leading coefficient of four.

We could also write this expression as four x squared plus zero x minus nine.

Now, important things to note are that we're going to need two numbers which we'll multiply to give us a negative nine.

And we need the same combination of numbers to give us a sum of zero.

This also means that to generate four x squared, we will need two identical terms in order to generate our value of zero for the coefficient of x.

Let's have a look at the factor pairs for negative nine.

We could choose one and negative nine, negative one and nine, or three and negative three.

As mentioned before, we needed two identical terms to multiply to give us four x squared.

So in this case it will be two x and two x.

And we need two numbers that will have a sum of zero.

So in our case, the only combination of numbers we could use will be three and negative three as this will give us six x minus six x.

So therefore, when we factorise this expression, we should get two x plus three and two x minus three.

Let's try this example.

So once again, we could rewrite this as nine x squared plus zero x minus 49.

So we need two numbers with a product of negative 49, and we need two identical terms which we'll multiply to give us nine x squared.

Let's have a look at the factor pairs of negative 49.

So we could use one and negative 49, negative one and 49, or seven and negative seven.

Our two identical terms, which we'll multiply to give us nine x squared will be three x.

And the only combination of numbers which would generate the sum of zero would be seven and negative seven.

So therefore, when we factorise this expression, we should get three x plus seven multiplied by three x minus seven.

Here's a question for you to try.

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

And here is your solution.

So when we look at this carefully we can see that when we factorise this, we needed two terms which were identical, which were two x and two x which we're given in the question.

And in our brackets, we have positive five and negative five as when we multiply these together they will give us our constant of negative 25.

Let's try this example.

So we need two identical terms which will give us a value of 16 x squared.

So two identical terms which we'll multiply together to give a 16 x squared would be four x.

We also know because we have a negative constant that we have one positive value and one negative value.

And we also know to generate the sum of zero, we would need two have the same number.

So in our case, this will be the number six, as six multiplied by negative six would give us negative 36.

Here's some questions for you to try.

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

And here are your answers.

If we look at question 2F, we can see that in the given expression our constant came first, and this changes what our brackets look like.

So in our solution, we get six plus seven x and six minus seven x.

This is because when we multiplied positive seven x and negative seven x, we needed to generate negative 49 x squared.

Here's another the question for you to try.

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

And here are the solutions.

So if we look at the first statement, we can see that we needed four x in each of our brackets to generate 16 x squared when we expand.

When we look at the second example, it looks like an attempt to half the coefficient of x squared has been made, so that's where 32 x possibly came from, but we actually need eight x as again, we're going to multiply eight x by eight x to generate 64 x squared.

When we look at the third example, we have nine x squared plus 16.

We can't actually factorised this expression.

This is because we would need two numbers with a product of 16, and we'd need the same two numbers to combine to give us a sum of zero, which cannot work.

And when we look at our last example, this is true.

So the only thing that's changed is that the constant is first in each of the brackets.

And that brings us to the end of our lesson.

I hope you've enjoyed factorising with a difference of two squares, I will hopefully see you soon.