Algebra Complete Study Guide – 50 Solved Problems, Step-by-Step Explanations & Formula Sheet

Algebra — Complete Study Guide
Introduction to Algebra

1. What is Algebra?




Algebra is a branch of mathematics where we use letters or symbols (variables) to represent
unknown numbers.

It allows us to solve problems systematically and understand mathematical relationships.



Example:

x + 5 = 12

Solve for x → x = 7

Why Algebra?

Simplifies complex arithmetic problems

Helps in solving equations

Essential for higher mathematics, science, engineering




2. Key Terms in Algebra



Term Definition Example




Variable Symbol representing unknown value x, y, z

Constant Fixed number 2, 5, 10

Coefficient Number multiplying a variable 3x → 3

Expression Combination of variables & constants 2x + 5

Equation Statement that two expressions are equal 2x + 3 = 11

, Types of Equations & Operations




1. Types of Equations

1. Linear Equation: Highest power of variable = 1

Example: 2x + 3 = 11




2. Quadratic Equation: Highest power = 2

Example: x² – 5x + 6 = 0




3. Cubic Equation: Highest power = 3

Example: x³ – 3x² + 2x = 0




4. Inequalities: Equations with <, >, ≤, ≥

Example: 3x + 2 > 8




2. Basic Algebraic Operations

Addition/Subtraction: (x + 3) + (x – 2) = 2x + 1

Multiplication: 2x × 3x = 6x²

Division: (6x² ÷ 2x) = 3x

Expanding: (x + 2)(x + 3) = x² + 5x + 6