🧠 Algebra Cheat Sheet for SSC Exams

This cheat sheet includes all essential algebraic formulas, identities, and shortcuts that appear frequently in SSC CGL, CHSL, and other competitive exams.

📌 Basic Identities

1. Square of a Binomial

Formula:
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²

Example:
Find the value of (7 + 3)².
Solution:
(7 + 3)² = 7² + 2×7×3 + 3² = 49 + 42 + 9 = 100

2. Difference of Squares

Formula:
a² - b² = (a + b)(a - b)

Example:
Evaluate: 36² - 34²
Solution:
Use identity: (36 + 34)(36 - 34) = 70 × 2 = 140

3. Cube of a Binomial

Formula:
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a - b)³ = a³ - 3a²b + 3ab² - b³

Example:
Find (2 + 1)³
Solution:
= 8 + 12 + 6 + 1 = 27

4. Sum and Difference of Cubes

Formula:
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)

Example:
Evaluate 8³ - 3³
Solution:
= (8 - 3)(64 + 24 + 9) = 5 × 97 = 485

🔥 Special Shortcuts

5. Identity for `x + 1/x`

If x + 1/x = a, then:

x² + 1/x² = a² - 2
x³ + 1/x³ = a³ - 3a

Example:
If x + 1/x = 3, find x² + 1/x²
Solution:
= 3² - 2 = 9 - 2 = 7

6. Symmetric Equations

If x + 1/x = a, then:

x⁴ + 1/x⁴ = a⁴ - 4a² + 2

Example:
If x + 1/x = 2, find x⁴ + 1/x⁴
Solution:
= 2⁴ - 4×2² + 2 = 16 - 16 + 2 = 2

🧩 Factorization Formulas

a² + b² + 2ab = (a + b)²
a² + b² - 2ab = (a - b)²
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Example:
Factor: x³ + y³ + z³ - 3xyz
Solution:
Use identity above.

🧮 Quadratic Equation Basics

General form: ax² + bx + c = 0
Roots:
x = [-b ± √(b² - 4ac)] / 2a

Discriminant:

If D > 0: Real and distinct roots
If D = 0: Real and equal roots
If D < 0: Imaginary roots

Example:
Solve x² - 5x + 6 = 0
Solution:
x = [5 ± √(25 - 24)] / 2 = [5 ± 1]/2 → x = 3, 2

🎯 SSC-Level Trick

Q: If a + b = 10 and ab = 21, find a² + b²
Trick:
Use identity: a² + b² = (a + b)² - 2ab
→ = 100 - 42 = 58

📋 Pro-Tip Table

Identity	Shortcut Use
`(a + b)²`	Common in simplification
`a³ + b³`	Useful in factoring questions
`x + 1/x` types	Found in algebraic expression
`x² + y²` from `x + y, xy`	Via identity
Quadratic formula	MCQs in SSC CGL/CHSL

✅ Final Tip

Always try to recognize identities and avoid expansion unless necessary. SSC exams focus more on applying the right trick quickly rather than lengthy solving.

Stay smart, practice regularly, and crack algebra like a pro!

Algebra Cheat Sheet for SSC Exams