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Home >> Standard Identities & their applications >> (a - b)2 = a2 + b2 - 2ab >>

A - B Whole Square

(a + b)2 = a2 + b2 + 2ab (a - b)2 = a2 + b2 - 2ab a2 - b2 = (a + b) (a b) (x + a) (x + b) = x2 + x(a + b) + ab (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
(a + b)3 = a3 + b3 + 3ab(a + b) (a - b)3 = a3 - b3 - 3ab(a - b) a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)

Before you understand A - B Whole Square : (a - b)2 = a2 + b2 - 2ab, you are advised to read:

How to Multiply Variables ?
How to Multiply Polynomials ?
What is Exponential Form ?

How this identity of (a - b)2 = a2 + b2 - 2ab is obtained:

Taking LHS of the identity:
(a - b)2

This can also be written as:
= (a - b) (a - b)

Multiply as we do multiplication of two binomials and we get:
= a(a - b) - b(a - b)
= a2 - ab - ab + b2

Add like terms and we get:
= a2 - 2ab + b2

Rearrange the terms and we get:
= a2 + b2 - 2ab

Hence, in this way we obtain the identity i.e. (a - b)2 = a2 + b2 - 2ab

Following are few applications this identity.


Example 1: Solve (3a - 2b)2
Solution: This proceeds as:
Given polynomial (3a - 2b)2 represents identity second i.e. (a - b)2
Where a = 3a and b = 2b

Now apply values of a and b on the identity i.e. (a - b)2 = a2 + b2 - 2ab and we get:
(3a - 2b)2 = (3a)2 + (2b)2 - 2(3a)(2b)

Expand the exponential forms and we get:
= 9a2 + 4b2 - 2(3a)(2b)

Solve multiplication process and we get:
= 9a2 + 4b2 - 12ab

Hence, (3a - 2b)2 = 9a2 + 4b2 - 12ab



Example 2: Solve (6m 9n)2
Solution: This proceeds as:
Given polynomial (6m 9n)2 represents identity second i.e. (a - b)2
Where a = 6m and b = 9n

Now apply values of a and b on the identity i.e. (a - b)2 = a2 + b2 - 2ab and we get:
(6m 9n)2 = (6m)2 + (9n)2 - 2(6m)(9n)

Expand the exponential forms and we get:
= 36m2 + 81n2 - 2(6m)(9n)

Solve multiplication process and we get:
= 36m2 + 81n2 - 108mn

Hence, (6m 9n)2 = 36m2 + 81n2 - 108mn



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