An algebraic identity is mathematical result of a calculation. They are shortcuts to speed up and simplify calculations which are commonly encountered.
For example:
(a+b)^2=a^2+2ab+b^2
(a+b)(a−b)=a^2−b^2

Algebraic Identities

Introduction:
Algebraic identities are mathematical formulas or equations that are true for all values of the variables involved. These identities help simplify complex algebraic expressions and solve equations more easily. They play a crucial role in algebra, providing a foundation for solving problems and proving other mathematical theorems.

Key Algebraic Identities:
There are several fundamental algebraic identities that are widely used in mathematics. Some of the key identities include:

1. Identity Property of Addition: a + 0 = a and 0 + a = a
This property states that adding zero to any number or adding any number to zero results in the same number.

2. Identity Property of Multiplication: a * 1 = a and 1 * a = a
This property states that multiplying any number by one or multiplying one by any number results in the same number.

3. Commutative Property of Addition: a + b = b + a
This property states that changing the order of the terms in an addition operation does not affect the sum.

4. Commutative Property of Multiplication: a * b = b * a
This property states that changing the order of the factors in a multiplication operation does not affect the product.

5. Associative Property of Addition: (a + b) + c = a + (b + c)
This property states that the grouping of terms in an addition operation does not affect the sum.

6. Associative Property of Multiplication: (a * b) * c = a * (b * c)
This property states that the grouping of factors in a multiplication operation does not affect the product.

Applications:
Algebraic identities are essential in various mathematical applications, including:

1. Simplifying Expressions: By applying algebraic identities, complex expressions can be simplified into more manageable forms. This simplification allows for easier calculations and analysis.

2. Solving Equations: Algebraic identities are used to transform equations into equivalent forms, making it easier to solve for unknown variables. These identities help in manipulating equations to isolate the desired variable.

3. Proving Theorems: Algebraic identities serve as a foundation for proving other mathematical theorems. By using these identities, mathematicians can establish relationships between different mathematical concepts and derive new results.

Conclusion:
Algebraic identities provide the basis for simplifying expressions, solving equations, and proving theorems in mathematics. They help in manipulating and transforming mathematical formulas, making complex problems more manageable. Understanding and applying these identities are essential skills for algebraic problem-solving and further advances in mathematics.