Algebra Identities – 90-Second Cheat Sheet
Algebra Identities – Quick Revision
Key Points (One-Liners)
- (a + b)² = a² + 2ab + b² – square of sum
- (a – b)² = a² – 2ab + b² – square of difference
- a² – b² = (a + b)(a – b) – difference of squares (most tested)
- (a + b)³ = a³ + b³ + 3ab(a + b) – cube of sum
- (a – b)³ = a³ – b³ – 3ab(a – b) – cube of difference
- a³ + b³ = (a + b)(a² – ab + b²) – sum of cubes
- a³ – b³ = (a – b)(a² + ab + b²) – difference of cubes
- (x + a)(x + b) = x² + (a + b)x + ab – product of two linear binomials
- If a + b + c = 0, then a² + b² + c² = –2(ab + bc + ca)
- a² + b² + c² – ab – bc – ca = ½[(a–b)² + (b–c)² + (c–a)²] ≥ 0
- (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) – square of trinomial
- a⁴ – b⁴ = (a² + b²)(a² – b²) = (a² + b²)(a + b)(a – b)
- Replace ‘a’ or ‘b’ with numbers to create quick MCQs
- Always factor first – cancelling reduces 50 % work
- Railway favours ‘difference of squares’ in every shift – master #3
- 30-second check: plug x = 0 or 1 to verify your identity
- Never expand fully in options – compare coefficients instead
- Keep factors ready: 1, 2, 3, 5, 7, 11 save division time
- Sign error in (a – b)² → biggest rank-loser; recheck twice
- Time-cap: 45 s per identity question – move on if stuck
| Formula/Rule |
Application |
| a² – b² = (a + b)(a – b) |
Simplification, surds, number-series |
| (a ± b)² = a² ± 2ab + b² |
Completing square, quadratic roots |
| (a + b + c)² |
Area/Perimeter problems with three variables |
| a³ ± b³ |
Volume/cube questions, linear equations |
| (x + a)(x + b) = x² + (a + b)x + ab |
Finding roots without quadratic formula |
| a⁴ – b⁴ |
Higher-order factors in simplification |
| If a + b = s, ab = p → a² + b² = s² – 2p |
Shortcut for symmetric sums |
| (a + b)³ expansion |
Often hidden in ‘find value’ type |
| a² + b² + c² – ab – bc – ca ≥ 0 |
Inequality/least-value questions |
| Replace variables with 1 to check options |
5-second elimination trick |
Memory Tricks
- SODA: Square Of Difference Always – a² – 2ab + b² (minus sign)
- PLUS-TWO: plus sign → 2ab; minus sign → –2ab
- “Cross-Bridge”: a² – b² = (a + b)(a – b) – imagine bridge halves
- Cube-Song: “Cube plus cube, plus three a b, times sum you see”
- Factor-Family: a³ + b³ → (a + b) is elder brother, always present
Common Mistakes
| Mistake |
Correct Approach |
| Writing (a – b)² = a² – b² |
Include middle term –2ab |
| Forgetting brackets in (a + b)³ |
Expand as (a + b)(a + b)(a + b) step-wise |
| Sign error in a³ – b³ factor |
Second factor starts with +ab |
| Cancelling a + b from both sides blindly |
Check if a + b = 0 first |
| Assuming (a + b)² = a² + b² |
Remember 2ab must be added |
Last Minute Tips
- Reach 15 min early – do two identity questions to warm up
- Start with ‘difference of squares’ questions – easiest & highest weight
- Use option substitution (x = 0, 1, –1) before expanding
- Keep 30 s hard-stop per question – mark & move, return later
- Darken bubble in one go; no partial shading in OMR
Quick Practice (5 MCQs)
Q1. If x² – y² = 24 and x – y = 4, then x + y equals
→ Use a² – b² = (a + b)(a – b) ⇒ 24 = (x + y)·4 ⇒ x + y = 6
Q2. The value of 98² – 2·98·2 + 2² is
→ (98 – 2)² = 96² = 9216
Q3. Simplify (a + b)³ – (a – b)³
→ 2b³ + 6ab² or 2b(b² + 3a²)
Q4. If a + b = 5 and ab = 6, find a² + b²
→ 5² – 2·6 = 25 – 12 = 13
Q5. The factors of x² – 7x + 12 are
→ (x – 3)(x – 4)
BETA
