a² − b² Formula

Formula: a² − b² = (a + b)(a − b)

Proof (by expansion): (a + b)(a − b) = a(a − b) + b(a − b) = a² − ab + ab − b² = a² − b² ✓

The middle terms −ab and +ab cancel out.

Examples:

1. x² − 9 = x² − 3² = (x + 3)(x − 3)

2. 25 − y² = 5² − y² = (5 + y)(5 − y)

3. 4x² − 49 = (2x)² − 7² = (2x + 7)(2x − 7)

4. 100 − 1 = 10² − 1² = (10 + 1)(10 − 1) = 11 × 9 = 99 ✓

The identity helps evaluate products quickly.

Method: ab = ((a+b)/2)² − ((a−b)/2)²

Example 1: 99 × 101 = (100 − 1)(100 + 1) = 100² − 1² = 10000 − 1 = 9999

Example 2: 48 × 52 = (50 − 2)(50 + 2) = 50² − 2² = 2500 − 4 = 2496

Example 3: 97 × 103 = (100 − 3)(100 + 3) = 100² − 3² = 10000 − 9 = 9991

This approach is faster than direct multiplication.

Standard algebraic identities:

1. (a + b)² = a² + 2ab + b²
2. (a − b)² = a² − 2ab + b²
3. a² − b² = (a + b)(a − b) ← this identity
4. (a + b)³ = a³ + 3a²b + 3ab² + b³
5. (a − b)³ = a³ − 3a²b + 3ab² − b³
6. a³ + b³ = (a + b)(a² − ab + b²)
7. a³ − b³ = (a − b)(a² + ab + b²)

Note: a² + b² cannot be factorised over real numbers. (It factors as (a + bi)(a − bi) over complex numbers only.)

Condition for applying a² − b²: Both terms must be perfect squares and the operation must be subtraction.