Expansions
Introduction
Algebraic expansions are the foundation of higher mathematics. ICSE Class 9 requires mastery of standard identities and their applications in simplifying expressions and performing quick numerical calculations.
Standard Identities
Identity 1: (a + b)²
(a + b)² = a² + 2ab + b²
Proof: (a + b)(a + b) = a² + ab + ab + b² = a² + 2ab + b²
<ICSEExample title="Expand (3x + 5y)²"> <Solution> (3x + 5y)² = (3x)² + 2(3x)(5y) + (5y)² = 9x² + 30xy + 25y² </Solution> </ICSEExample>Identity 2: (a — b)²
(a — b)² = a² — 2ab + b²
<ICSEExample title="Expand (4a — 7b)²"> <Solution> (4a — 7b)² = (4a)² — 2(4a)(7b) + (7b)² = 16a² — 56ab + 49b² </Solution> </ICSEExample>Identity 3: (a + b)(a — b)
(a + b)(a — b) = a² — b²
This is called the 'difference of squares' identity.
<ICSEExample title="Expand (5p + 3)(5p — 3)"> <Solution> (5p + 3)(5p — 3) = (5p)² — 3² = 25p² — 9 </Solution> </ICSEExample>Identity 4: (x + a)(x + b)
(x + a)(x + b) = x² + (a + b)x + ab
<ICSEExample title="Expand (x + 4)(x + 7)"> <Solution> (x + 4)(x + 7) = x² + (4 + 7)x + 4×7 = x² + 11x + 28 </Solution> </ICSEExample> <ICSEExample title="Expand (p — 3)(p + 8)"> <Solution> (p — 3)(p + 8) = p² + (-3 + 8)p + (-3)(8) = p² + 5p — 24 </Solution> </ICSEExample>Identity 5: (a + b)³
(a + b)³ = a³ + 3a²b + 3ab² + b³
Alternative form: (a + b)³ = a³ + b³ + 3ab(a + b)
<ICSEExample title="Expand (2x + 3y)³"> <Solution> (2x + 3y)³ = (2x)³ + 3(2x)²(3y) + 3(2x)(3y)² + (3y)³ = 8x³ + 3(4x²)(3y) + 3(2x)(9y²) + 27y³ = 8x³ + 36x²y + 54xy² + 27y³ </Solution> </ICSEExample>Identity 6: (a — b)³
(a — b)³ = a³ — 3a²b + 3ab² — b³
Alternative form: (a — b)³ = a³ — b³ — 3ab(a — b)
<ICSEExample title="Expand (2x — y)³"> <Solution> (2x — y)³ = (2x)³ — 3(2x)²(y) + 3(2x)(y)² — (y)³ = 8x³ — 12x²y + 6xy² — y³ </Solution> </ICSEExample>Identity 7: (a + b + c)²
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
<ICSEExample title="Expand (2x + 3y + 4z)²"> <Solution> (2x + 3y + 4z)² = (2x)² + (3y)² + (4z)² + 2(2x)(3y) + 2(3y)(4z) + 2(4z)(2x) = 4x² + 9y² + 16z² + 12xy + 24yz + 16zx </Solution> </ICSEExample>Applications in Numerical Calculation
The identities can be used to compute squares and products quickly.
<ICSEExample title="Evaluate 104² without direct multiplication"> <Solution> 104² = (100 + 4)² = 100² + 2×100×4 + 4² = 10000 + 800 + 16 = 10816 </Solution> </ICSEExample> <ICSEExample title="Evaluate 97²"> <Solution> 97² = (100 — 3)² = 100² — 2×100×3 + 3² = 10000 — 600 + 9 = 9409 </Solution> </ICSEExample> <ICSEExample title="Evaluate 103 × 97"> <Solution> 103 × 97 = (100 + 3)(100 — 3) = 100² — 3² = 10000 — 9 = 9991 </Solution> </ICSEExample>Common Mistakes With Fixes
| Mistake | Correction |
|---|---|
| Writing (a + b)² = a² + b² | Always include the middle term 2ab |
| Forgetting the 2 in cross terms for (a+b+c)² | (a+b+c)² = a²+b²+c² + 2(ab+bc+ca) |
| Error in (a — b)³ sign pattern | Signs alternate: +, -, +, - |
| Confusing (a+b)² with a²+b² | (a+b)² is always greater than a²+b² (by 2ab) |
ICSE Exam Focus
| Topic | Marks (approx.) | Frequency |
|---|---|---|
| Direct expansion using identities | 3 marks | Very common |
| Numerical evaluation using identities | 3 marks | Very common |
| Simplification of expressions | 4 marks | Common |
| Application of (a+b+c)² | 3 marks | Occasionally asked |
Self-Test
Q1: Expand: (i) (5a + 2b)² (ii) (3x — 4y)² (iii) (a + 2)(a — 5)
Q2: Evaluate using identity: (i) 105² (ii) 98² (iii) 102 × 96
Q3: Expand: (x + 2y + 3z)²
Q4: If a + b = 7 and ab = 10, find a² + b².
Q5: Expand (3a + 2b)³.
