An identity is an equality that holds true regardless of the values chosen for its variables.
For example, the identity
is always true regardless of the values of and .
Since identities are true for all valid values of its variables, one side of the equality can be swapped for the other. For example, we can replace any instance of with and vice versa because is an identity.
Clever use of identities offers shortcuts to many problems by making the algebra easier to manipulate. Below are lists of some common algebraic identities.
These identities are product formulas that are basic examples of the binomial theorem.
These identities are factoring formulas. Their more general forms are listed on the factorization page.
Application and Extensions
The identity holds for all real values of . What is ?
Multiplying out the left side of the identity we get
This expression must be equal to the right hand side of the identity, therefore
so , , and , which gives us .
If and , what is ?
While you could solve for and , a more elegant solution exploits the identity
which can be rewritten as
Substituting in and for and we get