Mr Barton’s Maths Notes
Algebra
5. Double Brackets
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5. Double Brackets
You knew it was coming…
Just when you have got your head around how to expand single brackets, your lovely maths
teacher announces it’s time to have a go at expanding double brackets.
But the good news is that it’s no more difficult than single brackets, you don’t need to learn
any new skills, and you get loads more marks for doing it!
Skills you need for success…
If you know about these things, you will be fine:
How to expand single brackets (see Algebra 2. Single Brackets)
Rules of Algebra (see Algebra 1. Rules of Algebra)
Rules of Negative Numbers (see Number 8. Negative Numbers)
•
•
•
It’s all about FOIL…
Now, like with most things in maths, there are a lot of different ways of expanding double
brackets, and if you are happy with your way, then just stick to it, but here is how I do it.
FOIL basically tells me the order in which I need to multiply terms, because the most
common mistake people make when expanding double brackets is to miss a few terms out!
1

(
Some people call
this the smiley face
method!
4
2

) (
)
3
1.
F irst
Multiply together the first terms in each bracket – remembering to
include the signs in front of them
2.
O uter
Multiply together the terms on the outside each bracket – remembering
to include the signs in front of them
3.
I nner
Multiply together the terms on the inside each bracket – remembering to
include the signs in front of them
4.
L ast
Multiply together the last terms in each bracket – remembering to
include the signs in front of them
Example 1
Example 2
(a  6) ( a  4)
Until you get really comfortable, there is
nothing wrong with drawing the smiley face on
to remind you what to multiply!
( p  10) ( p  8)
Time for the smiley face…
( p  10) ( p  8)
(a  6) ( a  4)
Be really careful with the NEGATIVES…
a  a
a  4
6a
6 4
First
Outer
Inner
Last




a2
4a
6a
24
Now we write down our answers, in order,
remembering if there is no sign in front of our
term it’s just a disguised plus!
a  4a  6a  24
2
Notice that the middle terms simplify to give…
a 2  10a  24
First
Outer
Inner
Last
p  p  p2
p  8  8p
10  p  10 p
10   8   80
Now we write down our answers, in order,
making sure we get all the signs correct!
p 2  8 p  10 p  80
Notice that the middle terms simplify to give…
p 2  2 p  80
Example 3
Example 4
(t  9) (t  2)
Let’s draw our smiley face…
(m  7) ( m  9)
Time for another smiley face…
(m  7) ( m  9)
(t  9) (t  2)
Again, we must watch those NEGATIVES…
First
Outer
Inner
Last
t  t  t2
t  2  2t
9  t   9t
9  2  18
Once again, the signs are the key to success!
t 2  2t  9t  18
Carefully simplify the middle terms…
t 2  7t  18
Be so, so, so careful with the NEGATIVES…
First
Outer
Inner
Last
m  m  m2
m   9   9m
7  m   7m
7   9  63
Writing down our answers, we get…
m 2  9m  7m  63
You have to know your Rules of Negative
Numbers inside out for this next bit…
m2  16m  63
Let’s take a moment to reflect…
Just before we look at a few more difficult ones (which, by the way, follow the exact same
rules as these), I just want to draw your attention to the answers we got…
(a  6) ( a  4)
a 2  10a  24
( p  10) ( p  8)
p 2  2 p  80
(t  9) (t  2)
t 2  7t  18
(m  7) ( m  9)
m2  16m  63
Now, look at the numbers in the questions and the numbers in the answers.
Can you see a quick way of getting from one to the other?...
Don’t worry if you can’t, but if you can then you are one step ahead, because that is the key
to success at 6. More Factorising, which is coming up soon…
But for now, how about some tricky expanding double bracket questions?...
Example 5
Example 6
(5 g  9) ( g  3)
Let’s draw our smiley face…
(3c  4) (2c  5)
Are you still feeling happy?…
(5 g  9) ( g  3)
Again, we must watch those NEGATIVES,
and we must know our Rules of Algebra!
First
5g
5g
9
9
Outer
Inner
Last




g
3
g
3




2
5g
15 g
 9g
 27
As always, the signs are the key to success!
5g  15 g  9 g  27
2
Carefully simplify the middle terms…
5 g 2  6 g  27
(3c  4) (2c  5)
NEGATIVES and Rules of Algebra again…
First
Outer
Inner
Last
3c  2c  6c 2
3c   5  15c
4  2c   8c
4   5  20
Writing down our answers, we get…
6c 2  15c  8c  20
Carefully simplify the middle terms…
6c 2  23c  20
Example 7
( a  b) (c  d )
Let’s draw our smiley face…
( a  b) (c  d )
Again, we must watch those NEGATIVES,
and we must know our Rules of Algebra!
First
Outer
Inner
Last
a  c  ac
a   d   ad
b  c  bc
b   d   bd
As always, the signs are the key to success!
ac  ad  bc  bd
Can we simplify the middle two (or indeed, any) of
the terms?... NO because there are NO LIKE
TERMS!
Example 8 – because I am feeling nasty…
(7ab  3bc) (5a 2  2c)
Are you still smiling now?…
(7ab  3bc) (5a 2  2c)
Okay, you would be really unlucky to ever get one
as hard as this, but there’s no reason we can’t do it
First
Outer
Inner
Last
7 ab  5a 2  35a 3b
7ab   2c  14abc
3bc  5a 2  15a 2bc
3bc   2c   6bc 2
Phew! Writing down our answers, we get…
35a 3b  14abc  15a 2bc  6bc 2
Can we simplify the middle two (or indeed, any) of
the terms?... NO because there are NO LIKE
TERMS!
Last one, I promise…
How would you do this one?...
(a  7) 2
If you said: “well, it’s dead easy, isn’t it, the answer is just…
a 2  49
Then please never say that again… because it’s wrong!
Remember: squaring something means multiplying it by itself.
So, this question could actually be written as…
(a  7) ( a  7)
Which means we can go back to our friend FOIL, and everyone is happy!
Incidentally, if you want to check you can still do these, the final simplified answer is…
a 2  14a  49
Can you see how we could have reached that answer a quicker way?…
TO BE CONTINUED on a maths website near you…
Good luck with
your revision!