Algebraic Identities & Their Uses

🎯 Learning Objective

Apply (a+b)², (a−b)², a²−b², (a+b)³ and a³+b³ to evaluate expressions and large squares fast.

💡 Concept

• (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².
• a² − b² = (a + b)(a − b) — the difference of squares.
• (a + b)³ = a³ + b³ + 3ab(a + b); similarly (a − b)³ = a³ − b³ − 3ab(a − b).
• a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).
• Handy result: a² + b² = (a + b)² − 2ab; and if a + 1/a = k then a² + 1/a² = k² − 2.

🧮 Key Formulas

(a ± b)² = a² ± 2ab + b²

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a² − b² = (a + b)(a − b)

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(a + b)³ = a³ + b³ + 3ab(a + b)

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a³ + b³ = (a + b)(a² − ab + b²)

✏️ Easy Example

Q. Evaluate (103)² using an identity.

1. Write 103 = 100 + 3, so (100 + 3)²
2. = 100² + 2(100)(3) + 3²
3. = 10000 + 600 + 9

Answer: 10609

🇮🇳 Real-Life Example

Mentally squaring 103 or 98 at a shop counter to estimate a bill uses (a±b)² — identities turn scary multiplications into two-step mental maths.

📝 Exam-Level Example

Q. If a + b = 7 and ab = 12, find a² + b².

1. a² + b² = (a + b)² − 2ab
2. = 7² − 2(12)
3. = 49 − 24

Answer: 25

📝 Exam-Level Example

Q. If a + b = 6 and ab = 8, find a³ + b³.

1. a³ + b³ = (a + b)³ − 3ab(a + b)
2. = 6³ − 3(8)(6)
3. = 216 − 144

Answer: 72

🪄 Memory Trick

When a value like x + 1/x is given, the answer for x² + 1/x² is just (that value)² − 2. No need to find x itself.

⚠️ Common Mistakes

• ❌ Forgetting the middle term 2ab in (a + b)²
• ❌ Writing a² − b² as (a − b)² by mistake
• ❌ Sign slip in (a − b)³ (the 3ab term is subtracted)

🏆 Exam Tips

• ✅ Break numbers near round figures: 97 = 100 − 3, 53² − 47² = (53+47)(53−47)
• ✅ Memorise a² + b² = (a+b)² − 2ab — it appears every year

📌 Summary

• (a ± b)² = a² ± 2ab + b²
• a² − b² = (a+b)(a−b)
• (a+b)³ = a³ + b³ + 3ab(a+b)
• a² + b² = (a+b)² − 2ab; x²+1/x² = (x+1/x)² − 2