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Algebra — Identities

Quantitative Aptitude · Algebra & Data Interpretation

Useful for: SSC CGL

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Overview

A few identities convert long expansions into one-line answers.

Key points

(a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; a² − b² = (a + b)(a − b).
(a + b)³ = a³ + b³ + 3ab(a + b); a³ + b³ = (a + b)(a² − ab + b²).
If a + b + c = 0, then a³ + b³ + c³ = 3abc.
x + 1/x = k → x² + 1/x² = k² − 2 → x³ + 1/x³ = k³ − 3k.
a² + b² = (a + b)² − 2ab.

Explanation

Substitute simple values (like a = 1, b = 0) to test options when stuck.
Symmetric expressions in a and b can be rewritten using a + b and ab only.
a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca).
For x − 1/x = k: x² + 1/x² = k² + 2.

Step by step

Identify the identity.
Substitute.
Simplify and verify with numbers.

Examples

x + 1/x = 3 → x² + 1/x² = 9 − 2 = 7 and x³ + 1/x³ = 27 − 9 = 18.
a + b = 7, ab = 10 → a² + b² = 49 − 20 = 29.
101² − 99² = (101 + 99)(101 − 99) = 400.

Traps and advanced points

Trap: Writing (a + b)² as a² + b².
Trap: Using k² + 2 for x + 1/x (it is k² − 2).
Trap: Forgetting the 3ab(a + b) term in a cube.

Quick revision

k² − 2 and k³ − 3k.
a + b + c = 0 → 3abc.
Test options with easy numbers.

Practice MCQs

If x + 1/x = 4, then x^2 + 1/x^2 equals:
1. 14
2. 16
3. 18
4. 12
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Answer: (a) 14. (x+1/x)^2 = x^2 + 1/x^2 + 2.
If a + b + c = 0, then a^3 + b^3 + c^3 equals:
1. 0
2. abc
3. a+b+c
4. 3abc
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Answer: (d) 3abc. Identity: a^3+b^3+c^3-3abc = (a+b+c)(…).
The factorised form of a^2 - b^2 is:
1. (a-b)(a-b)
2. (a+b)(a-b)
3. (a-b)^2
4. (a+b)^2
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Answer: (b) (a+b)(a-b). Difference of squares: a^2-b^2 = (a+b)(a-b).
If x - 1/x = 3, then x^2 + 1/x^2 equals:
1. 13
2. 11
3. 9
4. 7
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Answer: (b) 11. (x-1/x)^2 = x^2 + 1/x^2 - 2.
If x + 1/x = 2, then x^3 + 1/x^3 equals:
1. 4
2. 6
3. 8
4. 2
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Answer: (d) 2. x^3+1/x^3 = k^3 - 3k where k=2? — 8 - 6 = 2.
The value of (a+b)^2 - (a-b)^2 is:
1. 2(a^2+b^2)
2. 4a^2
3. 4ab
4. 2ab
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Answer: (c) 4ab. (a+b)^2 = a^2+2ab+b^2; (a-b)^2 = a^2-2ab+b^2.
If a^2 + b^2 = 29 and ab = 10, then (a+b) equals:
1. 6
2. 8
3. 7
4. 5
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Answer: (c) 7. (a+b)^2 = a^2+b^2+2ab = 29 + 20 = 49 → a+b = 7.
If 3^x = 81, then the value of x is:
1. 2
2. 4
3. 3
4. 5
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Answer: (b) 4. 81 = 3^4.
The simplest value of a^3 + b^3 when a = 2, b = 3 is:
1. 25
2. 40
3. 35
4. 30
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Answer: (c) 35. A^3+b^3 = 8 + 27 = 35.
If x + y = 12 and xy = 27, the value of x^2 + y^2 is:
1. 72
2. 90
3. 80
4. 100
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Answer: (b) 90. x^2+y^2 = (x+y)^2 - 2xy = 144 - 54 = 90.

Algebra — Identities

Overview

Key points

Explanation

Step by step

Examples

Traps and advanced points

Quick revision

Practice MCQs

More topics in this chapter