The three binomial formulas – basics
Formulas, derivation and practical calculation tricks
The three binomial formulas are fundamental identities of algebra that simplify multiplying out squared expressions. They hold for all real numbers a and b and are indispensable in school, university and work. The formulas are: (1) (a + b)² = a² + 2ab + b², (2) (a − b)² = a² − 2ab + b² and (3) (a + b)(a − b) = a² − b².
The derivation of the first binomial formula follows from direct multiplication: (a + b)² = (a + b)(a + b) = a² + ab + ab + b² = a² + 2ab + b². The "double product" 2ab arises because the mixed term ab appears twice. The second formula (a − b)² = (a − b)(a − b) likewise gives a² − ab − ab + b² = a² − 2ab + b². Here the double product is negative.
The third binomial formula is called the "difference of squares": (a + b)(a − b) = a² − ab + ab − b² = a² − b². The mixed terms cancel out. This formula is especially useful for mental math tricks: 49 × 51 = (50−1)(50+1) = 2500 − 1 = 2499. Or 97 × 103 = (100−3)(100+3) = 10000 − 9 = 9991. This calculation is noticeably faster than conventional written multiplication.
In analysis and algebra, the binomial formulas are used to simplify expressions, to complete the square (rewriting as a perfect square), and to solve quadratic equations. Extending them to arbitrary powers leads to the binomial theorem(a + b)ⁿ, where the coefficients follow Pascal's triangle. For n = 3: (a + b)³ = a³ + 3a²b + 3ab² + b³.