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Algebraic Identities & Equations
Mathematics · Algebra, Geometry & Data
Useful for: Railway RRB, CTET, OTET, OSSTET, UGC NET, OSSC RHT — Regular High School Teacher, UPSC NDA/NA, UPSC CDS CSIR NET, CTSA / STSS, Central Univ./IIT/NIT Faculty, Junior Teacher, OAVS, OPSC Assistant Professor, OPSC PGT, OSEPA Primary Teacher / JT Category-I, OSEPA Upper Primary — Science, OSSC Mathematics, OSSC Teacher Readiness, OSSSC Govt Physical Education Teacher (PET), OSSSC Govt TGT Arts, OSSSC Govt TGT Science — CBZ, OSSSC Govt TGT Science — PCM, Sainik Schools, Special OTET, SSB Lecturer / PGT, SSB Lecturer (Degree Colleges), SSB School Teachers, SSB TGT PCMand 21 more exams
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
For x² − 47x + 280 = 0, what is the sum of its roots?
- 49
- 51
- 45
- 47
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Answer: (d) 47. For ax²+bx+c=0, sum of roots = −b/a.
For x² − 58x + 216 = 0, what is the sum of its roots?
- 60
- 62
- 56
- 58
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Answer: (d) 58. For ax²+bx+c=0, sum of roots = −b/a.
For x² − 69x + 918 = 0, what is the sum of its roots?
- 67
- 71
- 73
- 69
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Answer: (d) 69. For ax²+bx+c=0, sum of roots = −b/a.
For x² − 7x + 12 = 0, what is the sum of its roots?
- 7
- 9
- 5
- 11
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Answer: (a) 7. For ax²+bx+c=0, sum of roots = −b/a.
For x² − 90x + 261 = 0, what is the sum of its roots?
- 88
- 90
- 92
- 94
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Answer: (b) 90. For ax²+bx+c=0, sum of roots = −b/a.
For x² − 17x + 72 = 0, what is the sum of its roots?
- 19
- 15
- 21
- 17
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Answer: (d) 17. For ax²+bx+c=0, sum of roots = −b/a.
Solve: 5x + 12 = 32.
- 4
- 8
- 6
- 2
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Answer: (a) 4. Correct answer: 4. Apply the core rule for Use identities such as (a+b)^2, (a−b)^2 and a^2−b^2.
For x² − 19x + 90 = 0, what is the sum of its roots?
- 17
- 21
- 19
- 23
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Answer: (c) 19. For ax²+bx+c=0, sum of roots = −b/a.
Solve: 3x + 7 = 64.
- 23
- 17
- 21
- 19
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Answer: (d) 19. Correct answer: 19. Apply the core rule for Use identities such as (a+b)^2, (a−b)^2 and a^2−b^2.
For x² − 21x + 110 = 0, what is the sum of its roots?
- 25
- 21
- 19
- 23
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Answer: (b) 21. For ax²+bx+c=0, sum of roots = −b/a.
Solve: 7x + 13 = 132.
- 21
- 19
- 17
- 15
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Answer: (c) 17. Correct answer: 17. Apply the core rule for Use identities such as (a+b)^2, (a−b)^2 and a^2−b^2.
For x² − 23x + 132 = 0, what is the sum of its roots?
- 27
- 21
- 25
- 23
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Answer: (d) 23. For ax²+bx+c=0, sum of roots = −b/a.
Solve: 6x + 6 = 66.
- 12
- 8
- 10
- 14
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Answer: (c) 10. 6x=60.
Solve: 3x + 6 = 54.
- 16
- 18
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- 14
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Answer: (a) 16. 3x=48.
Solve: 5x + 18 = 103.
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- 19
- 15
- 17
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Answer: (d) 17. 5x=85.
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