Special Products and Algebraic Identities: Formulas with

Some products appear so often that mathematicians gave them names and memorised formulas. (a + b)² always expands to a² + 2ab + b², and (a + b)(a − b) always gives a² − b². These special products turn slow multiplication into instant answers.

Identities are the shortcuts of algebra: they speed up mental arithmetic (103² = 10609 in seconds), unlock factorisation, and appear in nearly every chapter from quadratics to trigonometry. Exams reward students who recognise these patterns on sight.

Key Concepts

• Identity: an equation true for all values of its variables, unlike a conditional equation such as x + 2 = 5.
• Square of a sum: (a + b)² = a² + 2ab + b².
• Square of a difference: (a − b)² = a² − 2ab + b².
• Difference of squares: (a + b)(a − b) = a² − b².
• Product with a common term: (x + a)(x + b) = x² + (a + b)x + ab.
• Cube identities: (a + b)³ = a³ + 3a²b + 3ab² + b³ and a³ − b³ = (a − b)(a² + ab + b²).

Formulas

• (a + b)² = a² + 2ab + b² — square both parts, then add twice the product.
• (a − b)² = a² − 2ab + b² — same, but subtract twice the product.
• (a + b)(a − b) = a² − b² — the middle terms cancel completely.
• (x + a)(x + b) = x² + (a + b)x + ab — the x-coefficient is the sum, the constant is the product.
• a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).
• All identities hold for all real numbers a, b, x.

Worked Example

Compute 103² without a calculator, using an identity.

1. Rewrite: 103 = 100 + 3, so 103² = (100 + 3)².
2. Apply (a + b)² = a² + 2ab + b² with a = 100, b = 3.
3. a² = 10,000; 2ab = 2 × 100 × 3 = 600; b² = 9.
4. Add: 10,000 + 600 + 9 = 10,609.

103² = 10,609. Verify by ordinary multiplication: 103 × 103 = 103 × 100 + 103 × 3 = 10,300 + 309 = 10,609. Matches. Similarly, 97² = (100 − 3)² = 10,000 − 600 + 9 = 9,409.

Common Misconceptions

• "(a + b)² = a² + b²." The most famous algebra mistake in history. The missing 2ab term matters: (2 + 3)² = 25, but 2² + 3² = 13. Never equal unless ab = 0.
• "(a − b)² = a² − b²." Also wrong: (a − b)² = a² − 2ab + b². Test with numbers: (5 − 2)² = 9, while 25 − 4 = 21.
• "Identities only work for small nice numbers." They hold for every value — fractions, negatives, decimals, even other expressions: (2x + 5y)² = 4x² + 20xy + 25y².

Real-World Applications

• Mental-math champions square numbers near 100 using these identities.
• Factorisation — the reverse of expansion — solves quadratic equations.
• Physics simplifies expressions like (v + u)(v − u) = v² − u².
• Geometry uses (a + b)² to expand areas of enlarged squares.

Special products turn repeated multiplications into memorised patterns: square the parts, double the cross term, watch the signs. Learn the five core identities cold, prove each once with FOIL, and test the classic mistakes with numbers so you never make them.