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➧ Actually, in an identity the expressions on both side of an 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.(a + b)2 = a2 + b2 + 2ab
Proof:LHS
= (a + b)2
= (a + b)(a + b)
= a(a + b) + b(a + b)
= aa + ab + ba + bb
= a2 + 2ab + b2
= RHS
(a − b)2 = a2 + b2 − 2ab
Proof:LHS
= (a − b)2
= (a − b)(a − b)
= a(a − b) − b(a − b)
= aa − ab − ba + bb
= a2 −2ab + b2
= RHS
(a + b)(a − b) = a2 − b2
Proof:LHS
= (a+b)(a − b)
= a(a − b) + b(a − b)
= aa − ab + ba − bb
= a2 − b2
= RHS
(x + a)(x + b) = x2 + (a + b)x + ab
Proof:LHS
= (x + a)(x + b)
= x(x + b) + a(x + b)
= xx + bx + ax + ab
= x2 + (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
= x2+ 32+ 2⋅x⋅3
{∵(a + b)2 = a2 + b2 + 2ab }
= x2+ 9 +6x
297 × 303
= (300−3)(300+3)
= (300+3)(300−3)
= (300) 2 - 32
{∵(a+b)(a − b) = a2 − b2}
= 90000 - 9
= 89991
Finding Squares
(b – 7)2= b2 −2⋅b⋅7 + 72
= b2 −14b + 49
9982
= (1000−2)2
= 10002 −2⋅1000⋅2 + 22
= 1000000 −4000 + 4
= 996000 + 4
= 996004
Simplifying
(ab +bc)2– 2ab2c= (ab)2+ (bc)2+ 2abbc – 2ab2c
= a2b2+ b2c2+ 2ab2c – 2ab2c
= a2b2+ b2c2
| Simple Equation | |