Algebraic identities

Equations that are always true

An equation like 2x + 3 = 11 is true only when x = 4. An algebraic identity is true for every value:

(a + b)² = a² + 2ab + b²

Try a = 3, b = 4: (7)² = 49 and 9 + 24 + 16 = 49 ✓. It works for any numbers.

The key identities

The first two give perfect square trinomials.

Why (a + b)² isn’t a² + b²

Picture a square with side (a + b). It splits into:

The total is a² + 2ab + b². The two "ab" rectangles are what people forget.

Using identities to expand

To expand a binomial squared, use the pattern.

• (x + 5)² = x² + 10x + 25
• (2y − 3)² = 4y² − 12y + 9
• (m + 7)(m − 7) = m² − 49

Using identities to factor

To factor, recognize the pattern in reverse.

• x² − 36 = (x + 6)(x − 6)
• 9a² − 25 = (3a + 5)(3a − 5)
• x² + 14x + 49 = (x + 7)²
• 4p² − 20p + 25 = (2p − 5)²

Test for a perfect square trinomial: first and last terms are perfect squares, and the middle term is twice the product of their roots.

Mental maths tricks

• 51² = (50 + 1)² = 2,500 + 100 + 1 = 2,601
• 49 × 51 = (50 − 1)(50 + 1) = 2,500 − 1 = 2,499
• 98² = (100 − 2)² = 10,000 − 400 + 4 = 9,604

Identity vs equation

Worked examples

Example 1: Expand (3x + 2)² → 9x² + 12x + 4.

Example 2: Factor 16 − y² → (4 + y)(4 − y).

Example 3: Is x² + 6x + 8 a perfect square? Last term 8 isn’t a square → no.

Common mistakes

• (a + b)² = a² + b² (forgetting 2ab).
• Sign errors: (a − b)² has −2ab but +b².
• Trying to factor a sum of squares like x² + 9 with real numbers. It doesn’t factor.

Try it yourself

1. Expand (x − 4)².
2. Expand (5a + b)(5a − b).
3. Factor x² − 81.
4. Factor 25x² + 30x + 9.
5. Use an identity to calculate 103².

Answers: 1) x² − 8x + 16 2) 25a² − b² 3) (x + 9)(x − 9) 4) (5x + 3)² 5) 10,609

Quick recap

• An algebraic identity is true for all values.
• Key identities: (a ± b)² and the difference of squares.
• Use them to expand, factor and calculate mentally.