How to: expand brackets

25 04 2010

So you’re looking at the exam paper and you see a question that looks something like this:

Expand 3(2x + 5)

You’re thinking to yourself, ‘what do I do?’ and ‘what does expand mean?’

Don’t worry, it’s not difficult, all you need to know is that the word ‘expand’ just means do some multiplication. When you expand the brackets in the question above all you are doing is some multiplication. In my explanation below I’m going to use ‘X’ to mean ‘mulitiply’ or ‘times and ‘x’ to mean the letter ‘x’.

Lets start off by thinking about what 3(2x + 5) actually means. In algebra, mathematicians are a bit lazy and we don’t write in the multiplication signs. Instead of writing 3 X a we’d just write 3a. When we look at 3(2x + 5) what it actually means is 3 multiplied by 2x + 5. To do this multiplication all you have to do is draw out a grid like this:

Notice how I put the 3 which is outside the brackets on the left hand side. I also split up the two terms inside the brackets, the 2x and the +5 and put them in separate columns on the top row. All you do now is you fill in the empty places in the grid by multiplying the numbers in the row and column together. The first space is filled by 3 X 2x which is 6x and the second space is filled by 3 X +5 which is + 15. Notice how I kept the plus sign in front of the 15. That helps us remember that this is positive and not negative. So our grid now looks like this:

To finish off, now all we do is to add together all the things that we got from doing the multiplications. In our case this is 6x + 15. So:

Expand 3(2x + 5) = 6x + 15

That’s it! Not so difficult after all! Let’s have a look at another one:

Expand 2x(3x – 7)

The first thing we do is put the question into our grid:

Notice how I have kept the negative sign, the ‘-‘ infront of the 7. When you do the 2x X -7 we have to remember that we are multiplying by negative 7. So, fill in the grid by doing the multiplications and it will look like this:

Remember that when we multiplied the 2x by 3x that we do the numbers first, 2 X 3 to get the 6 and then we do the letters x times x is x². Then -7 times 2x is -14x. Now we just add together all the things that we got from the mulitplications:

Expand 2x(3x – 7) = 6x² – 14x

Hopefully you’ll have got the hang of this by now and you can see how you can answer this type of question. They are all done the same way and the three important things you need to remember are:

  1. Don’t be scared by the word ‘expand’. All it means is that you have to do some multiplication.
  2. Put the terms into a multiplication grid. Put the bit outside the bracket down the left hand side and split the terms in the bracket up and put them in two columns at the top.
  3. Once you’ve filled in the grid by doing the multiplications, just add together all the results and that’s the answer to the question!

There’s one more type of question you might get on expanding brackets. These are solved in a very similar way so don’t panic! Sometimes you might get a question where you have to multiply two brackets together. It might look like this:

Expand (2x + 7)(3x – 2)

This just means 2x + 7 multiplied by 3x – 2. We solve it by using the same grid method. You split the two terms in the first bracket into 2x and +7 and put them down the left hand side of the grid in separate rows like shown below. You then split the two terms in the second bracket into 3x and -2 and put them in separate columns at the top. The multiplication grid will look like this:

Notice now how we have four little multiplications to do. Once we’ve done them our grid will look like this:

Remember to put in the + and – signs when you are doing the multiplication because they are really important when we come to next bit which is adding up all our answers. When we add up what we’ve got we have:

6x² + 21x – 4x -14

We can simplify this by collecting together how many x we have. We’ve got 21 of them and then we take 4 away so actually we have 17x. E.g. 21x – 4x = 17x. So our answer is actually:

Expand (2x + 7)(3x – 2) = 6x² + 17x -14

That’s all there is to it! This is all you need to know to answer any questions about multiplying brackets. I hope I’ve made it nice and clear for you. To get really good at it you just need to do lots of practice.

Use the comments section below to share your own practice questions and answers and to get help from other people.

Good luck!

P.S. to multiply two brackets together you might have heard of the FOIL method rather than the grid method that I’ve used above. If you can use the FOIL method reliably then do so, that’s totally fine. I encourage pupils to use the grid method because I have found that they make less mistakes.

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