Transcript Introduction to Factorising Quadratics and

Creating brackets
In this powerpoint, we meet 5 different methods
of factorising.
Type 1 – Common Factor
This involves taking
a term outside the
brackets. Always
try to do this first.
Type 2 – Difference of Two
Squares
Try this when you
have two terms with
a minus between
Type 3 – Grouping
This is the easiest
one to pick – use it
when there are 4
terms!
Types 4 and 5
Use these for expressions
with 3 terms.
Quadratic trinomials
They will be of the format
x2 + bx + c (Type 4) OR
ax2 + bx + c (Type 5)
Where a, b and c are just numbers
Factorising just makes me
sooooo happy!!
Summary
Type
When to Use
1. Common
factor

2. Difference of
Two squares

3. Grouping





4. Quadratic
Trinomial (I)
5. Quadratic
Trinomial (II)


Always try first before any other method
Examples: a2 – 9a ; 2xy + 5x2
When there are only 2 terms which are squares
There must be a minus sign
Examples: a2 – 25 ; 81 – 4b2 ; w4 – 16
There are 4 terms.
Example: a2 – 4a + 3ab – 12b
There are 3 terms. Has a squared term.
Examples: a2 – 9a + 20 ; 6 – 5b + b2
There are 3 terms. Has a squared term with a
number attached in front.
 Examples: 2a2 – 3a – 5 ; 6b – 5b2 + 3b

Type 1 of 5 – common factor
Always try this first, regardless
Always look
of what type it is
for a common
factor!
3a – 12 =
3a2 – 12a =
20ab – 12b2 =
30a6 – 15a5 =
3a2 + 6a + 12 =
3(a – 4)
3a(a – 4)
4b(5a – 3b)
15a5(2a – 1)
3(a2 + 2a + 4)
Remember – take out the largest factor you can!
Type 2 of 5 – diff of 2 squares
To qualify as a Type 2, an
expression
• must have only 2 terms which are SQUARES
• must have a MINUS sign separating them
Examples
a2 – 9 = (a – 3)(a + 3)
16 – a2 = (4 – a)(4 + a)
(2b)2 – (3a)2 = (2b – 3a)(2b + 3a)
9b2 – 25 = (3b – 5)(3b + 5)
Combining Types 1 and 2
Example 1 .....Factorise 5x2 – 45
LookMum !
It’s a
difference
of 2
squares!
STEP 1
Treat as a Type 1, and take
out common factor first, 5
Write 5(x2 – 9)
STEP 2
Now do expression in brackets as a
Type 2
Write 5(x
– 3)(x + 3)...ANS!
Example 2 .....Factorise x4 – 81
STEP 1
Treat as a Type 2, and write as
difference of 2 squares.....
(x2 – 9)(x2 + 9)
STEP 2
Now check out the thing in each
bracket. We can factorise the first
one, but not the second.
(x2 – 9)(x2 + 9)
(x – 3)(x + 3)(x2 + 9)....ANS!!
Y’can’t factorise a SUM of two
squares Stupid! x2 + 9 has to
stay as it is. It’s not the same
as (x + 3)(x + 3) is it now???
Example 3 .....Factorise 80a4 – 405b12
STEP 1
Identify common factor, 5 and remove
Write 5(16a4 – 81b12)
STEP 2
Now work on the terms in the brackets
This is a difference of 2 squares and
becomes (4a2 – 9b6) (4a2 + 9b6)
Write
STEP 3
5(4a2 – 9b6) (4a2 + 9b6)
Now work on the terms in the 1st bracket.
This is a difference of 2 squares and
becomes (2a – 3b3) (2a + 3b3) . Write
5(2a – 3b3) (2a + 3b3) (4a2 + 9b6)
Example 4 .....Factorise 9a2 – (x – 2a)2
Just treat as difference of 2 squares of the format
9a2 – b2
where the b = [x – 2a]
Factorising it then becomes
= (3a – b)(3a + b)
And then replacing the b with [x – 2a] we get
= (3a – [x – 2a])(3a + [x – 2a])
Now get rid of square brackets
= (3a – x + 2a)(3a + x – 2a)
Clean up
= (5a – x )(a + x)
Ans!!
You could check your answer by expanding it and also expanding
the original question. They should both give the same thing.
Type 3 of 5 – Grouping
You can tell when you’ve got one of these because
there are FOUR TERMS !!!
Example 1
Factorise 2a – 4b + ax – 2bx
STEP 1 – split it into “2 by 2”
= 2a – 4b
+ ax – 2bx
No need to be
confused!
STEP 2 – factorise each pair separately as Type 1
= 2(a – 2b)
+ x(a – 2b)
STEP 3 – take out the (a – 2b) as a common factor
= (a – 2b)(2 + x)...ans!!
Type 3 of 5 – Grouping
Example 2
If these are
the same,
it’s a good
sign!
Factorise xy + 5x – 2y – 10
STEP 1 – split it into “2 by 2”
= xy + 5x
– 2y – 10
STEP 2 – factorise each pair separately as Type 1
= x(y + 5)
– 2 (y + 5)
STEP 3 – take out the (y + 5) as a factor
= (y + 5)(x – 2) ans!!
Type 3 of 5 – Grouping
Example 3
Factorise x2 – x – 5x + 5
STEP 1 – split it into “2 by 2”
= x2 – x
– 5x + 5
Ewbewdy!!
They’re the same!
 On my way to a VHA

STEP 2 – factorise each pair separately as Type 1
= x(x – 1)
– 5 (x – 1)
STEP 3 – take out the (x – 1) as a
factor
= (x – 1 )(x – 5) ans!!
Example 4 - harder
Factorise x2 – 4y2 – 2ax – 4ay
STEP 1 – split it into “2 by 2”
= x2 – 4y2
– 2ax – 4ay
Awwright!
They’re the
same!!
STEP 2 – factorise each pair separately
1st pair – Type 2
= (x – 2y) (x + 2y)
2nd pair – Type 1
– 2a (x + 2y)
STEP 3 – take out the (x + 2y) as a factor
= (x + 2y)(x – 2y – 2a) ans!!
Type 4 of 5 –
Easy Quadratic Trinomial
You can usually pick these as they have 3 TERMS
Example 1 .....Factorise x2 + 5x + 6
STEP 1 – Make 2 brackets
(x..............)(x.............)
STEP 2 – Look for 2 numbers that
Add to make +5
Multiply to make +6
STEP 3 – Put ‘em in the brackets
+2 &
+3
(x + 2)(x + 3)
Type 4 of 5 –
Easy Quadratic Trinomial
Example 2 .....Factorise 2x2 – 6x – 20
STEP 1 – take out a common factor (remember
this should be your 1st step EVERY time!!)
= 2(x2 – 3x – 10)
STEP 2 – Ignore the 2. For the expression inside
the brackets, look for 2 numbers that
Add to make – 3
Multiply to make – 10
+2 & – 5
STEP 3 – Put ‘em in the brackets
2(x + 2)(x – 5)
Type 4 of 5 –
Easy Quadratic Trinomial
Example 3 .....Factorise 6 + 5x – x2
STEP 1 – Rearrange into “normal” format with
x2 at the front, then x, then the number
= – x2 + 5x + 6
STEP 2 – Now take out a common factor – 1
= – (x2 – 5x – 6)
STEP 3 – Ignore the minus. Look for 2 numbers
that add to – 5, and multiply to – 6.
These are +1 and –6.
– (x + 1)(x – 6)
Type 5 of 5 –
Harder Quadratic Trinomial
With a number
in front of the
x2
Example 1 .....Factorise 2x2 + 5x – 3
STEP 1 – Draw up a fraction like this
(2x ........)(2x ........)
2
STEP 2 – Look for two numbers that
ADD to make +5
MULT to make – 6
2×–3=–6
(2x  6)(2x  1) Numbers are +6, – 1

2
Note the 2 in bottom must
= (x + 3)(2x – 1) ANS
cancel one whole bracket
FULLY! So (2x + 6) becomes
(x + 3)
Type 5 of 5 –
Harder Quadratic Trinomial
With a number
in front of the
x2
Example 2 .....Factorise 3x2 + 8x – 3
STEP 1 – Draw up a fraction like this
(3x ........)(3x ........)
3
STEP 2 – Look for two numbers that
ADD to make +8
MULT to make – 9
3×–3=–9
(3x  9)(3x  1) Numbers are +9, – 1

3
Note the 3 in bottom must
= (x + 3)(3x – 1) ANS
cancel one whole bracket
FULLY! So (3x + 9) becomes
(x + 3)
Type 5 of 5 –
Harder Quadratic Trinomial
With a number
in front of the
x2
Example 3 .....Factorise 6x2 – 19x + 10
STEP 1 – Draw up a fraction like this
(6x ........)(6x ........)
6
STEP 2 – Look for two numbers that
ADD to make –19
MULT to make 60
6 × 10 = 60
(6x  4)(6x  15) Numbers are –4 , –15

23
Note the 6 in bottom would
= (3x – 2)(2x – 5) ANS
not cancel either bracket
FULLY! So we broke the 6
into 2 x 3 then cancelled.
Now wozn’t
that just a
barrel of
fun??