Exploring Algebraic Identities

An algebraic identity is an equation that holds true for all possible values of its variables. For example, (a + b)² = a² + 2ab + b² is an identity because it is correct no matter what numbers a and b stand for. Identities are used to expand and simplify expressions quickly.

The most common identities are (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and a² − b² = (a + b)(a − b). Class 9 also extends these to expressions with three terms and to cubes such as (a + b)³.

Recognise the form of the expression and match it to a known identity. For example, x² − 9 matches a² − b² with a = x and b = 3, so it factorises as (x + 3)(x − 3). Matching the pattern avoids long multiplication.

An identity is true for every value of the variable, while an ordinary equation is true only for particular values. For instance, (a + b)² = a² + 2ab + b² is an identity, but x + 2 = 5 is an equation true only when x = 3.

Identities make calculations and simplifications faster and reduce errors. They let us expand brackets, factorise expressions, and even compute products like 102 × 98 mentally by writing them as (100 + 2)(100 − 2). They are a key tool throughout algebra.

Use the identity (a + b)² = a² + 2ab + b². Square the first term, add twice the product of the two terms, then add the square of the second term. For example, (x + 5)² = x² + 10x + 25.