# Factorisation Using Identities

## Formula

• (a + b)2 = a2 + 2ab + b2
• (a - b)2 = a2 - 2ab + b2
• (a + b)(a - b) = a2 - b2

## Notes

### Factorisation Using Identities:

• (a + b)2 = a2 + 2ab + b2
• (a - b)2 = a2 - 2ab + b2
• (a + b)(a - b) = a2 - b2
Factorise x2 + 8x + 16
Solution:

a2 + 2ab + b2 =  x2 + 2(x)(4) + 42 = x2 + 8x + 16

a2 + 2ab + b2 = (a + b)2

x2 + 8x + 16 = (x + 4)2        .....(the required factorisation)

## Example

Factorise 4y2 – 12y + 9
4y2 = (2y)2, 9 = 32 and 12y = 2 × 3 × (2y)
Therefore,
4y2 – 12y + 9
= (2y)2 – 2 × 3 × (2y) + (3)2
=( 2y – 3)2        .........(required factorisation)

## Example

Factorise 49p2 – 36

49p2 - 36
= (7p)2 - (6)2

= (7p - 6)(7p + 6)     .......(required factorisation)

## Example

Factorise a2 – 2ab + b2 – c2.

a2 - 2ab + b2 - c2
= (a - b)2 - c2
= [(a - b) - c)((a - b) + c)]
= (a - b - c)(a - b + c)        ......(required factorisation)

## Example

Factorise m4 – 256

m4 = (m2)2 and 256 = (16)2
m4 – 256
= (m2)2 – (16)2
= (m2 – 16) (m2 + 16)
Now,
(m2 + 16) cannot be factorised further, but (m2 – 16) is factorisable again as (a + b)(a - b) = a2 - b2
m2 – 16 = m2 – 42 = (m – 4)(m + 4)
Therefore,
m4 – 256 = (m – 4)(m + 4)(m2 + 16)
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