Algebraic Identities – A Reference for Common Patterns

An algebraic identity is an equation that's true for every value of its variables, not just some – unlike a normal equation, which is only true for specific solutions. The three identities you've been building toward across the last few pages are worth collecting together as a single reference: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and a² − b² = (a + b)(a − b).

These three patterns show up constantly throughout algebra, and recognising them on sight – rather than re-deriving them every time – is what separates confident algebra from slow, mechanical algebra. From here on, you can treat these as tools to reach for immediately whenever an expression matches their shape.

The Three Standard Identities

Real-Life Application

• Mental arithmetic tricks: squaring numbers near a round number quickly, using the identities.
• Engineering formulas: many formulas are pre-simplified using these identities.
• Computer science: identities help simplify expressions before writing efficient code.

Key Takeaways

Practice: Algebraic Identities