Lesson video

Hello, My name is Mr. Clasper and today, we're going to learn how to factorise a quadratic with a difference of two squares.

Let's begin with the following statement.

We've been told that A squared subtract B squared will always equal A plus B, all multiplied by A subtract B.

Let's take a look at this further.

If we take a square with a side of A, that square would have an area of A squared.

Likewise, if we took a different square with a side of B, this square would have an area of B squared.

We can show A squared minus B squared by taking the square of A and subtracting the square of B from it.

We now have some new dimensions.

So we have one side which is labelled as A minus B.

This is because originally, the full length was A, and we subtracted the length of B from it, therefore, this length must be A minus B.

Now, we can split the shape into two rectangles.

And we can take the smaller rectangle and align it to the side of the larger rectangle.

This leaves us with a rectangle of sides A plus B and A minus B.

Therefore, the area must be A plus B multiplied by A minus B, proving that that is also equal to A squared minus B squared.

Let's factorise X squared minus 16.

Now, this example might look slightly different to ones that you've tried in the past.

Another way to think about this is to think of it as X squared plus 0X minus 16.

Or in other words, we're looking for two numbers, which have a product of -16 and the sum of 0.

Let's use a multiplication grid to help us.

If we look at factor pairs for the number -16, we could choose 1 and -16 or -1 and 16, and we could also choose a -4 and 4.

Let's try 1 and -16.

So when try 1 and -16, we get X and -16X, which would have a sum of negative 15X.

However, we want a sum of 0, so these two are not going to work.

If we choose 4 and -4, we get positive for 4X and -4X, which do have a sum of 0, leaving us with the expression X squared minus 16.

Therefore, to factorise it, we get X plus 4 and X minus 4, a difference of two squares.

Let's factorise X squared minus 81.

For this example, we need two numbers with a product of negative 81, however, we need the same two numbers to have a sum of 0.

So we could use 1 and -81 or -1 or 81, but neither of these have a sum of 0.

We could try 3 and -27 or -3 and 27, but again, these two do not have a sum of 0.

If we use 9 and -9, these do have a sum of 0, therefore, when we factorise, we should get X plus 9 and X minus 9.

Here's a question for you to try.

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

And here are your answers.

Let's factorise X squared minus 36.

Remember, this can be written as X squared plus 0X minus 36.

This means we're looking for two numbers with a product of -36 and a sum of 0.

So within our bracket, we need an X in each 'cause when we multiply these we'll get X squared.

We know that we need a positive value and the negative value as this is the only way to create a negative constant when we expand them.

And we need the same two numbers in order to generate a sum of 0.

So in this case, that would be 6.

That means that X squared minus 36 must always be equal to X plus 6 and X minus 6.

Here are some questions for you to try.

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

And if you look carefully at question two f, you will notice that the constant is first in our expression which changed our answer, so our answer was 5 plus X and 5 subtract X.

Here's another question for you to try.

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

And here are your answers.

So these are the only three expressions which are equivalent to one another.

And that's it for today's lesson.

Thank you very much.

You take care.