Master NIMCET: Algebra - Mathematics

Algebra - NIMCET Preparation

Introduction

Algebra is the foundation of many mathematical concepts and is key to understanding more advanced topics. It focuses on expressions, equations, inequalities, and functions. Mastering algebra helps in solving complex problems in various areas like calculus, geometry, and statistics.

Algebraic Expressions

An algebraic expression is a combination of variables, constants, and operators. The expression can be simplified, expanded, or factored based on given rules.

Example: Simplify the expression (3x + 2y) - (2x - 4y).
Solution: The simplified expression is x + 6y.

Linear Equations

Linear equations are equations of the first degree, meaning they involve variables raised to the power of one. They can be solved using methods like substitution, elimination, or graphical representation.

General form: ax + by = c
Example: Solve the system of equations:
2x + 3y = 6
x - y = 1
Solution: The solution is x = 3, y = 1.

Quadratic Equations

Quadratic equations involve variables raised to the power of two. The standard form is:

ax² + bx + c = 0

The solutions to a quadratic equation can be found using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a
Example: Solve the quadratic equation x² - 5x + 6 = 0.
Solution: The roots are x = 2 and x = 3.

Inequalities

Inequalities express a relationship where one value is greater than, less than, or not equal to another. Solving inequalities involves similar steps as solving equations, but the inequality sign must be carefully handled when multiplying or dividing by negative numbers.

Example: Solve 2x - 3 > 7.
Solution: x > 5

Functions

Functions describe a relationship between two variables, typically written as f(x). Common functions include linear, quadratic, and exponential functions.

Linear function: y = mx + c
Quadratic function: y = ax² + bx + c

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