Mr F

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Mr Fs Maths Notes

• Algebra
• 6. Factorising Quadratics

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6. More Factorising Quadratics
Again, you knew it was coming Just like we had
to expand double brackets, it should come as no
surprise that we have to factorise expressions
back into double brackets as well! There is a
bit of a trick to this, and to discover it, lets
look back at our answers from 5.
Expanding Brackets
Note These answers are called Quadratic
Expressions because they have a squared term in
them
Focus your attention on the numbers if you can
see how to get from the numbers in the questions
to the numbers in the answers then you should
be able to see how to get from the answer back to
the question and if you can do that, then you
can already factorise quadratic expressions!
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How to Factorise Quadratic Expressions Factorising
quadratics means you want to get from
to
To be able to do this you need to be able to
solve a little puzzle If you look back at the
examples, you will see that
In other words, the two numbers in the bracket
(including their sign) must ADD TOGETHER to
give you the number (and sign) in front of the
x And MULTIPLY TOGETHER to give you the
number (and sign) at the end

x
So if you can discover what two numbers solve
that little puzzle, then you can factorise
quadratics and practice makes perfect!
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Example 1
Example 2
Okay, here the question we must ask ourselves
Okay, here the question we must ask ourselves
Which two numbers multiply together to give 24
and add together to give 11?
Which two numbers multiply together to give
-15 and add together to give 2?
Now, if you find it helps, you can write down all
the pairs of numbers which multiply together to
give 24, and see which one also adds up to 11
Again, nothing wrong with writing down pairs that
multiply together to give -15, but be careful of
your negatives!
x
x
1 x 24 1 24 25 2 x 12 2
12 14 3 x 8 3 8 11
1 x -15 1 -15 -14 -1 x
15 -1 15 14 3 x -5 3 -5
-2 -3 x 5 -3 5 2
x
x
v
x
v
Once we have our pair, we just write them in the
brackets, remembering that no sign is just a
disguised plus!
Once we have our pair, we just write the numbers
in the brackets, making sure we get our signs in
the correct place!
or
or
Why not expand the brackets to make doubly sure
you are correct!
Again, expanding the brackets is a good check!
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Example 3
Example 4
Okay, here the question we must ask ourselves
Okay, here the question we must ask ourselves
Which two numbers multiply together to give
-14 and add together to give -13?
Which two numbers multiply together to give 18
and add together to give -9?
Now, switch on your brain here we CANT be
talking two positive numbers, as how will they
add up to give -9?... so we need two negatives!
Unless you can do it in your head, just write
down pairs of numbers that multiply together to
give -14
-1 x -18 -1 -18 -19 -2 x
-9 -2 -9 -11 -3 x -6 -3
-6 -9
x
x
-1 x 14 -1 14 13 1 x -14 1
-14 -13
x
v
v
Once we have our pair, we just write the numbers
in the brackets, making sure we get our signs in
the correct place!
People tend to mess these up, but we wont,
because we know that two negatives multiplied
together gives a positive!
or
or
If you expand the brackets you will definitely
know that you are correct!
Again, expanding the brackets is a good check!
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What abut this funny looking one?
Okay, looks a bit strange, but lets ask
ourselves the same question as we always do
Which two numbers multiply together to give -16
and add together to give ermwell erm 0?
Remember, it is the number in front of the x
which tells us what the numbers must add together
to make, but we dont have any xs, so the sum of
our two numbers must be 0! Think of the
expression like this is it helps
So, isnt it true that for two numbers to add
together to give zero, they must be the same
number, but of opposite sign, so they cancel each
other out! So, which two numbers do we need?... 4
and -4! Expand it to check!
Expressions like this are called the difference
of two squares, and are always factorised in a
similar way. Look at these three examples and see
if you can see how I got the answers
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When things get a little tricky
Okay, whilst it was not so tricky to spot how to
factorise those types of quadratics, what about
when there is a number in front of the squared
term?... Lets look back at two examples we did
in 5. Expanding Double Brackets, to see if we can
spot where the numbers come from
Any ideas?... Its not easy to spot, is it? I
think the best thing I can do is to try and take
you through two examples as carefully as I
can Are you ready?....
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Example 1
Okay, lets start by thinking what the first
terms in the two brackets must be?... Do you
agree that they would have to be 2x and x,
otherwise we would not get out 2x2! How about the
numbers at the end of each bracket?... Well, they
would have to be a pair of numbers which multiply
together to give -3! I set out this information
in a table like this
All the pairs of numbers which multiply to give -3
2x -1 3 1 -3
x 3 -1 -3 1
The first terms in the brackets
Next I multiply DIAGONALLY, and I am looking for
a pair of numbers which will add up to the amount
of xs I need which from the question is -1x!
x
(2x x 3) (x x -1) 6x -1x
5x (2x x -1) (x x 3) -2x 3x
1x (2x x 1) (x x -3) 2x
-3x -1x
2x -1 3 -3 1
x 3 -1 1 -3
x
v
The pair I want is the 3rd column of numbers
along Now, to get my answer, I just put the two
terms from the top row in the first bracket, and
the two terms from the second row in my second
bracket
You have done so much work here, that it is
definitely worth checking you are correct by
expanding the brackets!
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Example 2
Okay, again lets start by thinking what the
first terms in the two brackets must
be?... Problem They could be either 8x and x,
or the could be 4x and 2x. Both, when multiplied
together, would give us the 8x2 that we need! So
this time we need two tables!
All the pairs of numbers which multiply to give
-15
The first terms in the brackets
8x -1 15 1 -15 -3 5 3 -5
x 15 -1 -15 1 5 -3 -5 3
4x -1 15 1 -15 -3 5 3 -5
2x 15 -1 -15 1 5 -3 -5 3
And once again I must just keep I multiplying
DIAGONALLY (in my head if I can), looking for a
pair of numbers which will add up to the amount
of xs I need which from the question is -2x!
After a long search, I reckon the pair with the
rings around them is what I need!
(4x x -3) (2x x 5) -12x 10x
-2x
As before, to get my answer I just put the two
terms from the top row in the first bracket, and
the two terms from the second row in my second
bracket
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• Good luck with your revision!