# Quadratic factorisation – 7 methods – Step by step What is quadratic factorisation? And how can we factorise a quadratic expression?

Today we are going to see 7 basic methods to factorise a quadratic expression.

And these methods are what you need.

Firstly you need to know what a quadratic expression is.

A quadratic expression in an expression of the form ax² + bx +c where x is the variable, and a, b are constants with a ≠ 0. For example:   x² + 3x + 4,  4x² − 16 and 4x² − 4x + 1 are quadratic expressions.

You did study the expansion of algebraic factors, many of which resulted in quadratic expressions. And factorisation is the reverse process of expansion.

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Factorisation is the process of writing an expression as a product of factors.

You can say “factoring” or “factorising” as this process is finding what to multiply to get the expression (finding the factors).

For example: (x + 1) and (x + 3) are factors of x² + 4x + 3.

You should remember the following expansion rules: Expanding is usually easy but factoring can often be tricky. Sometimes we can guess and check. But sometimes it doesn’t work.

## #1 Quadratic Factorisation by removal of common factors

Some quadratic expressions can be factorised by removing the Highest Common Factor (HCF) of the terms in the expression.

You need to check if there are any common factors before proceeding with any other factorisation.

Example 1: Factorise by removing a common factor:

a) 6x² + 2x

You can see 6 and 2 have a common factor of 2.

And x² and x have a common factor of x.

Therefore, 6x² + 2x has a common factor of 2x.

Then, 6x² + 2x = 2x(3x + 1).

b) −2x² − 3x

−2x² − 3x  has a common factor of −x.

Thus, −2x² − 3x = −x(2x + 3)

−x(2x + 3) = −x.2x − x.3 = −2x² − 3 ✅✅✅ OK!

Example 2: Fully factorise by removing a common factor:

Let’s check if there are any common factors.

a) (x − 3)² − 3(x − 3)

= (x − 3) (x − 3) − 3(x − 3)     <—- common factor = (x − 3)

=  (x − 3) [(x − 3) − 3]

= (x − 3)(x − 6)                     <— simplifying

b) (x + 2)² + 3x + 6

= (x + 2)(x+2) + 3(x + 2)    <— find a common factor by factoring 3x + 6

= (x + 2)(x + 2+ 3)

= (x + 2)(x + 5)

## #2-3 Quadratic Factorisation by using difference of two squares or perfect square expansion

a² − b ² = (a − b)(a + b)

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

Note: The sum of two squares does not factorise into two real linear fators.

(a + b)² and (a − b)² are perfect squares!

Example 1: Use perfect square rules to fully factorise :

a) x² + 10x + 25

= x² + 2 × x × 5 + 5²

= (x + 5)²

b) x² − 12x + 36

= x² − 2 × x × 6 + 6²

= (x − 6)²

c) 9x² − 6x + 1

= (3x)² − 2 × 3x × 1 + 1²

= (3x − 1)²

d) − 8x² − 24x − 18

= −2 (4x² + 12x + 9)    <—- common factor = −2

= −2 [(2x)² + 2 × 2x × 3 + 3²]

= −2(2x + 3)²

e) m² −  20x + 100

= (m − 10)²

f) − x² + 2x − 1

= − (x² − 2x + 1)

= − (x − 1)²

How to solve quadratic equations – 3 methods – step by step

Example 2: Fully factories:

a) 11 − x²

= (√11)² − x²

= (√11 − x)(√11 + x)

b) (x − 3)² − 5

= (x − 3)² − (√5)²

= [(x + 3) + √5] [(x + 3) − √5]

= (x + 3 + √5)(x + 3 − √5)

c) (x + 2)² − (x − 1)²

= [(x + 2) + (x − 1)] [(x + 2) − (x − 1)]

= [ x + 2 + x − 1] [ x + 2 − x + 1]

= (2x + 1)(3)

= 3(2x + 1)

## #4. Quadratic Factorising expressions with four terms by grouping them in two pairs.

Sometimes we can factorise an expression containing four terms by grouping them in two pairs.

For example,

ax² + 2x + 2 + ax can be rewritten as

ax² + ax + 2x + 2     <<< putting terms contain a together

= ax(x + 1) + 2(x + 1)    <<< factorising each pair

= (x + 1)(ax + 2)    <<< (x + 1) is a common factor

Example: Fully factorise:

a) ax + by + bx + ay

= ax + ay + by + bx   <<< putting terms contain a together

= a(x + y) + b(y + x)  <<< factorising each pair

= (x + y)(a + b)        <<< (x + y) is a common factor

b) 2x² − 20 + 4x − 10x

= 2(x² − 10 + 2x − 5x)  <<< 2 is a common factor

= 2(x² + 2x − 5x − 10<<< splitting into two pairs

= 2[x(x + 2) − 5(x + 2)<<< factorising each pair

= 2[(x + 2)(x − 5)]     <<< (x + 2) is a common factor

= 2(x + 2)(x − 5)

## #5. Quadratic Factorisation by using Sum and product method

x² + (p + q)x + pq = (x + p)(x + q)

We see that the coefficient of x is the sum of p and q and the constant term is the product of p and q.

So, if we are asked to factorise x² + 7x + 6, we need to look for two numbers with a product of 6 and a sum of 7.

These numbers are 1 and 6, and so

x² + 7x + 6 = (x + 1)(x + 6).

Example: Fully factorise

a) x² + 5x + 4

x² + 5x + 4  has p + q = 5 and pq = 4

Therefore, p and q are 1 and 4.

We have x² + 5x + 4  = (x + 1)(x + 4)

b) x² − x − 12

x² − x − 12  has p + q = − 1 and pq = − 12.

It follows that p and q are −4 and 3.

So, x² − x − 12 = (x − 4)(x + 3)

c) −2x² + 2x + 12

= − 2( x² − x − 6)    <<< removing − 2 as a common factor

= − 2(x − 3)(x + 2)   <<< sum = −1 and product = − 6, so the numbers are − 3 and 2

d) 4x − x² + 32

= − x² + 4x + 32

= − 1( x² − 4x − 32) <<< removing − 1 as a common factor

= − (x² − 8x + 4x − 32) <<< sum = −4 and product = − 32, so the numbers are − 8 and 4

= − (x − 8)(x + 4)

## #6. Quadratic factorisation by using Miscellaneous factorisation

Use the following steps in order to factorise quadratic expressions:

Step 1: Look carefully at the quadratic expression, if there is a common factor, take it out.

Step 2: Look for a perfect square factorisation or difference of two squares

Step 3: Look for the sum and product type.

Example: Fully factorise the following expressions:

a) x² − 16x + 39

= x² − 2 × x × 8 + 64 − 25

= (x − 8)² − 25    <<< perfect square

= (x − 8)² − 5²    <<< difference of two squares

= (x − 8 + 5)(x − 8  − 5)

= (x − 3)(x − 13)

b) 4x² − 8x − 60

= 4(x² − 2x − 15)   <<< taking out the common factor

= 4(x − 5)(x + 3)  <<< product = − 15 and sum = − 2

c) − 2x² − 14x − 36

= − 2(x² + 7x + 12)  <<< taking out the common factor

We need to find two numbers with a product of 12 and a sum of 7.

These numbers are 3 and 4.

− 2(x² + 7x + 12)

= − 2(x +3)(x + 4)

(Quadratic factorisation by using Miscellaneous method)

## #7. Quadratic factorisation (a ≠ 1) by ‘splitting’ the x-term

Quadratic factorisation by “splitting” the x-term is used when factorising a quadratic expression which does not fall into any of these categories.

Use the following steps to factorise ax² + bx + c by “splitting” the x-term:

Step 1: Find ac and then the factors of ac which add to b.

Step 2: If these factors are p and q, replace bx by px + qx.

Step 3: Complete the factorisation

Example:

a) 3x² + 7x + 2

We have ac = 3 × 2 = 6 and b = 7.

We need 2 numbers with a product of 6 and a sum of 7.

Searching amongst the factors of 6, only 1 and  6 have a sum of 7.

These are 1 and 6, so the split is 7x = x + 6x

3x² + 7x + 2

= 3x² + x + 6x + 2

= x(3x + 1) + 2(3x + 1)    <<< factoring in pairs

= (3x + 1)(x + 2)            <<< taking out the common factor

b) 10x² − 23x − 5

We can see that ac = 10 × (-5) = −50 and b = − 23.

We need 2 numbers with a product of −50 and a sum of −23.

Searching amongst the factors of − 50, only −25 and 2 have a sum of − 23.

These are −25 and 2.

So, the split is − 23x = −25x + 2x

10x² − 23x − 5

= 10x² − 25x + 2x − 5

= 5x(2x − 5) + (2x − 5)  <<< factoring in pairs

= (2x − 5)(5x + 1)    <<< taking out the common factor

c) 3x² − x − 10

ac = 3 × (-10) = − 30 and b = − 1.

We need two numbers with a product of − 30 and a sum of − 1.

Searching amongst the factors of − 30, only 5 and − 6 have a sum of −1.

3x² − x − 10

= 3x² + 5x − 6x − 10     <<< splitting the x-term

= x(3x + 5) − 2(3x + 5)  <<< factoring in pairs

= (3x + 5)(x − 2)          <<< taking out the common factor

(Quadratic factorisation by “splitting” the x-term) – it’s done!

Remember to check your factorisations by expansion!

Now, you know what quadratic factorisation is and how to factorise a quadratic expression.

7 methods are:

1. Quadratic factorisation by removal of common factors
2. Difference of two squares factorisation
3. Perfect square factorisation
4. Quadratic factorisation with four terms by grouping them in two pairs.
5. Quadratic factorisation by using Sum and product method
6. Quadratic factorisation by using Miscellaneous factorisation
7. Quadratic factorisation (a ≠ 1) by ‘splitting’ the x-term 