Algebra, the cornerstone of mathematics, empowers us to solve problems, analyze relationships, and understand the world around us. By using symbolic variables to represent unknown quantities, it allows us to manipulate expressions and equations to extract valuable information. Let's embark on a journey through the core concepts of Basic Algebra:
1. Addition and Subtraction:
- Addition: Combining quantities, represented by the "+" symbol. Example: 3 + 5 = 8.
- Subtraction: Finding the difference between quantities, represented by the "-" symbol. Example: 10 - 4 = 6.
- Properties: Commutative (order doesn't matter): a + b = b + a; Associative (grouping doesn't matter): (a + b) + c = a + (b + c).
2. Multiplication Formulas:
- Basic Multiplication: Combining groups of equal items, represented by the "x" or "*" symbol. Example: 2 x 3 = 6 (2 groups of 3).
- Distributive Property: Distributing quantities over addition/subtraction: a(b + c) = ab + ac; (a - b)c = ac - bc.
- Identity Property: 1 multiplied by any number is itself: 1x = x.
- Zero Product Property: If a product is 0, at least one factor must be 0: a x b = 0 implies a = 0 or b = 0 (or both).
3. Multiplication of Algebraic Sums:
- Expanding Parentheses: Multiplying a factor outside of parentheses: (a + b)(c + d) = ac + ad + bc + bd.
- Special Products: Simplifying binomials/polynomials multiplied: (a + b)(a - b) = a^2 - b^2; (a + b)^2 = a^2 + 2ab + b^2.
4. Division:
- Basic Division: Splitting a quantity into equal groups, represented by the "/" or ":" symbol. Example: 12 ÷ 3 = 4 (12 divided into 3 equal groups).
- Fraction Division: Inverting the divisor and multiplying: a / b = a \times b^-1.
- Zero Division: Division by zero is undefined.
5. Important Relations and Formulas:
- Order of Operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
- Powers and Exponents: Repeated multiplication (a^n = a multiplied by itself n times). Properties: a^m \times a^n = a^(m + n), (a^m)^n = a^(m \times n).
- Square Root: Finding a number that, when multiplied by itself, equals a given number (a^2 = b).
- Inequalities: Comparing quantities (<, >, ≤, ≥).
6. Quadratic Equations:
- General Form: ax^2 + bx + c = 0, where a ≠ 0.
- Solving Methods:
- Factoring: Rearranging the equation to express it as a product of two linear expressions.
- Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a).
- Completing the Square: Transforming the equation into a perfect square trinomial.
7. Logarithms:
- Definition: The logarithm of b to base a (log_a b) is the exponent to which you raise a to get b (a^log_a b = b).
- Common Logarithms: log_10 b is written as log b (base 10).
- Logarithm Properties: log(ab) = log a + log b, log(a^n) = n log a, log_a a = 1.
8. Progressions:
- Arithmetic Progression (AP): A sequence of numbers where the difference between consecutive terms is constant (e.g., 2, 5, 8, 11...).
- Geometric Progression (GP): A sequence of numbers where the ratio between consecutive terms is constant (e.g., 2, 4, 8, 16...).
9. Arithmetical Mean (AM): The average of a set of numbers, calculated by summing the numbers and dividing by the number of numbers.
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Maths