Important Algebraic Identities

Basic Identities (Must Memorize!)

1. (a + b)² = a² + 2ab + b²

2. (a - b)² = a² - 2ab + b²

3. a² - b² = (a + b)(a - b)

4. (a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b)

5. (a - b)³ = a³ - 3a²b + 3ab² - b³ = a³ - b³ - 3ab(a - b)

6. a³ + b³ = (a + b)(a² - ab + b²)

7. a³ - b³ = (a - b)(a² + ab + b²)

8. (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

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

10. If a + b + c = 0, then a³ + b³ + c³ = 3abc

Advanced Identities

11. (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

12. a⁴ - b⁴ = (a - b)(a + b)(a² + b²)

13. a⁴ + a²b² + b⁴ = (a² + ab + b²)(a² - ab + b²)

Linear Equations

One Variable

ax + b = 0 x = -b/a

Two Variables (Simultaneous)

Method 1: Substitution Method 2: Elimination Method 3: Cross Multiplication

Quadratic Equations

Standard Form

ax² + bx + c = 0

Roots = [-b ± √(b² - 4ac)] / 2a

Sum and Product of Roots

Sum of roots (α + β) = -b/a Product of roots (αβ) = c/a

Nature of Roots

If b² - 4ac > 0 → Real and distinct roots If b² - 4ac = 0 → Real and equal roots If b² - 4ac < 0 → Imaginary roots

Shortcuts & Tricks

Shortcut 1: Quick Squaring

(a + b)² = (a - b)² + 4ab a² + b² = (a + b)² - 2ab a² + b² = (a - b)² + 2ab

Shortcut 2: Value Finding

If a + 1/a = 5, find a² + 1/a²

(a + 1/a)² = a² + 1/a² + 2 25 = a² + 1/a² + 2 a² + 1/a² = 23

Shortcut 3: Factorization Patterns

x² + (a+b)x + ab = (x+a)(x+b) Example: x² + 5x + 6 = (x+2)(x+3)

Solved Examples

Q1: If a + b = 10 and ab = 21, find a² + b²

Solution:

a² + b² = (a+b)² - 2ab = 10² - 2(21) = 100 - 42 = 58

Q2: Factorize: x² - 5x + 6

Solution:

Find two numbers: sum = -5, product = 6 Numbers: -2, -3 x² - 5x + 6 = (x-2)(x-3)

Q3: Solve: 2x + 3 = 11

Solution:

2x = 11 - 3 = 8 x = 4

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