Lesson video

Hi, I'm Mrs. Dennett.

And in today's lesson, we're going to be learning how to expand and simplify double brackets.

In this lesson, you may have find it useful to picture algebra tiles.

It might be helpful to recap how to use them before we start in case you haven't come across them before.

The blue tile has a length of X and a width of X, so it's area is X squared.

We use this tile to represent X squared.

The green rectangle has a length of X and a width of 1, so it's area is 1X or X.

The little yellow square has a length of 1 and a width of 1, so it's area is 1.

The red tiles represent -X squared, -X and -1.

If I want to represent X plus 1 and X plus 2, I can do this in a rectangle.

X plus 1 is one dimension, and X plus 2 is the other.

I can use the other tiles to fill in the area.

Sometimes we can get mixed up with the tiles used for dimensions.

So I'm going to get rid of those, and write X plus 1 to show the length, and X plus 2 to show the width.

We can also represent these two brackets with a multiplication grid like this.

We can use algebra tiles to help us expand and simplify double brackets, like these ones here.

The brackets are being multiplied together.

But just like when we write terms such as 5X, we omit the multiplication sign between 5 and X, but we know it means 5 times X.

So here, we're multiplying X plus 1 by X plus 4.

As the two expressions, X plus 1 and X plus 4 are being multiplied together, this is just like finding an area.

We can use algebra tiles to show the dimensions.

It doesn't matter which way around you do this as multiplication is communicative, it can be done in any order.

We can fill in the rectangle by matching up the dimensions.

X times X gives us X squared.

We need four Xs followed by one X at the bottom, And four 1 tiles.

We can also represent this expansion in a multiplication grid.

X times X gives us X squared.

X times 4 gives us 4X.

1 times X gives us X, and 1 times 4 is 4.

We can see when we remove the dimensions for clarity, that we end up with X squared plus 4X plus X plus 4.

This simplifies by collecting 4X and X to give us X squared plus 5X plus 4.

Here's a question for you to try.

Pause the video to complete the task, and restart when you finished.

Here's the answer.

In the grid, we get X squared, 3X, 2X and 6.

Collecting like terms gives us X squared plus 5X plus 6.

There is another method for expanding double brackets that you may like to try.

We are multiplying everything in the first bracket by everything in the second.

We multiply X by X and X by 4, giving us X squared plus 4X.

Then, we multiply 3 by X and 3 by 4 to give us 3X plus 12.

Simplifying the X terms gives us X squared plus 7X plus 12.

Here's a question for you to try.

Use whichever method you feel most comfortable with.

Accuracy is the most important thing.

Pause the video to complete the task and restart when you finished.

Here are the answers.

The only things to be aware of in this question are the negative numbers in some of the expressions.

Remember that multiplying a negative number by another negative number gives a positive result.

Secondly, the other way that letters and numbers can be in any position within the brackets, the letters don't always come first.

Here's another question for you to try.

Pause the video to complete the task and restart when you finished.

Here is Ayda's answer.

She missed out the middle X terms when she expanded the brackets.

To avoid this, when you have a single bracket being multiplied by itself, write out both brackets.

So really, Ayda should've written out X plus 8 in one bracket being multiplied by X plus 8 in another bracket.

This should help you to abide making the same mistake as Ayda has done.

It's important that we can apply our skills to other contexts.

Here's a question in the context of area.

Pause the video to complete the task and restart when you finished.

Here's the answer.

We multiply the length by the width, we put each expression in brackets.

So the width would be C minus 4 in a bracket, and this is multiplied by the length, 5 plus C in a bracket.

We get these terms 5C, C squared, 20 and minus 4C.

Yours may be in a different order, but this doesn't matter as long as you the correct signs.

Simplify 5C, takeaway 4C to get C.

And we get C squared plus C minus 20 as our final answer.

Here's a final question for you to try.

Pause the video to complete the task and restart when you finished Here are the answers.

Did you notice the we got a positive sign in one bracket and a negative sign in the other bracket? This means that the X terms often called the middle terms, because of where they appear in the expression, cancel each other out, leaving us with just two terms. These two terms consist of squared letter term, and a square number, but they takeaway in between.

For this reason, this type of expression is called a difference of two squares.

That's all for today's lesson.

Thank you very much for watching.