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An equality that is true for every value of the variables in it is called an identity.

➧ Actually, in an identity, expressions on both side of the equality are same but in different form, so they are always equal for different values of variables in it and hence the equality is called an identity.

➧ We can substitute the one expression of the identity for the other in any problem.

➧ An equation is true for only certain values of the variable in it i.e., the expressions in an equation are equal for certain values of variable only.

# Standard Identities

The following four identities are most common and most used identities in algebra and hence called standard identities.

Proof:

LHS

*= (a + b) ^{2}*

*= (a + b)(a + b)*

*= a(a + b) + b(a + b)*

*= aa + ab + ba + bb*

*= a ^{2} + 2ab + b^{2}*

= RHS

Proof:

LHS

*= (a − b) ^{2}*

*= (a − b)(a − b)*

*= a(a − b) − b(a − b)*

*= aa − ab − ba + bb*

*= a ^{2} −2ab + b^{2}*

= RHS

Proof:

LHS

*= (a+b)(a − b)*

*= a(a − b) + b(a − b)*

*= aa − ab + ba − bb*

*= a ^{2} − b^{2}*

= RHS

Proof:

LHS

*= (x + a)(x + b)*

*= x(x + b) + a(x + b)*

*= xx + bx + ax + ab*

*= x ^{2} + (a+b)x +ab*

= RHS

## Applications of Standard Identities

Standard identities are applied in simplifying and finding products and squares of numbers and algebraic expressions.

### Finding Products

(*x* + 3) (*x* + 3)

= (*x* + 3)^{2}

= *x*^{2}+ 3^{2}+ 2⋅*x*⋅3

{∵(*a + b*)^{2} = *a*^{2} + *b*^{2} + 2*ab *}

= *x*^{2} + 9 + 6*x*

297 × 303

= (300−3)(300+3)

= (300+3)(300−3)

= (300)^{2} – 3^{2}

{∵(*a*+*b*)(*a*−* b*)=*a*^{2}−* b*^{2}}

= 90000 – 9

= 89991

### Finding Squares

(*b* – 7)^{2}

= *b*^{2} −2⋅*b⋅*7 + 7^{2}

= *b*^{2} −14*b* + 49

998^{2}

= (1000−2)^{2}

= 1000^{2} −2⋅1000⋅2 + 2^{2}

= 1000000 −4000 + 4

= 996000 + 4

= 996004

### Simplifying

(*ab *+*bc*)^{2}– 2*ab*^{2}*c*

= (*ab)*^{2} + (*bc*)^{2} + 2*abbc *– 2*ab*2*c*

= *a*^{2}*b*^{2} + *b*^{2}*c*^{2} + 2*ab*^{2}*c*– 2*ab*^{2}*c*

= *a*^{2}*b*^{2}+ *b*^{2}*c*^{2}

⏪Simple Equation |
Factorisation & Division of Algebraic Expressions⏩ |