Algebraic Identities — (a±b)², (a+b)³, and More

• Memorise and apply Identity I: (a + b)² = a² + 2ab + b².
• Apply Identity II: (a − b)² = a² − 2ab + b².
• Use Identity III: (a + b)(a − b) = a² − b².
• Expand (a + b)³ and (a − b)³ using NCERT formulas.
• Evaluate products like (998)² and factorise using identities in reverse.

1. (a + b)² = a² + 2ab + b² — square of sum.
2. (a − b)² = a² − 2ab + b² — square of difference.
3. (a + b)(a − b) = a² − b² — difference of squares.
4. (a + b)³ = a³ + 3a²b + 3ab² + b³.
5. (a − b)³ = a³ − 3a²b + 3ab² − b³.
6. (x + a)(x + b) = x² + (a + b)x + ab — useful for quick multiplication.
7. Mental math — (99)² = (100 − 1)² = 9801.
8. Factorisation — Recognise a² − b², perfect square trinomials a² ± 2ab + b².
9. NCERT Ex. 2.5 — Expand (x + 2y + 4z)² using repeated identity (extension).
10. Common factor first — Always check for HCF before applying identities.

NCERT: Evaluate (104)² without direct multiplication

Use (a + b)² with a = 100, b = 4:
(104)² = (100 + 4)²
= 100² + 2(100)(4) + 4²
= 10000 + 800 + 16
= 10816
Tip: Choose a as nearest round number for fast calculation

• Writing (a + b)² = a² + b² (missing 2ab term).
• Sign error in (a − b)² — middle term is −2ab, not +2ab.
• Confusing (a + b)³ with a³ + b³ (need middle terms).
• Forgetting 2ab when a or b is negative: (−3 + x)² still has +2(−3)(x).
• Using a² − b² = (a − b)² (wrong — that is (a − b)² = a² − 2ab + b²).

• Expand (3x − 2)².
• Factorise x² − 49 using an identity.
• Evaluate (995)² using (1000 − 5)².
• Expand (2a + 3b)³ — write first two terms.
• Simplify (x + 5)(x − 5).

Open the Practice tab for graded questions on Algebraic Identities — (a±b)², (a+b)³, and More.