Algebra Concepts and Techniques

Definition: A branch of mathematics dealing with symbols and the rules for manipulating those symbols.
Expressions: Combinations of numbers, variables, and operators (e.g., (3x + 2)).
Equations: Statements that two expressions are equal (e.g., (2x + 3 = 7)).
Functions: A relation between a set of inputs and a set of permissible outputs, typically expressed as (f(x)).
Linear Equations: Equations of the first degree (e.g., (y = mx + b)).
Quadratic Equations: Equations of the second degree (e.g., (ax^2 + bx + c = 0)).
- Factoring: Process of breaking down an expression into simpler components (e.g., (x^2 - 5x + 6 = (x-2)(x-3))).
- Quadratic Formula: Used to find roots of quadratic equations: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
Systems of Equations: Sets of equations with multiple variables that can be solved simultaneously.
- Methods: Substitution, elimination, graphing.
Inequalities: Mathematical statements that describe a range of values (e.g., (x > 5)).
Polynomials: Expressions consisting of variables raised to whole number powers, combined using addition, subtraction, and multiplication.

Definition: A measure of central tendency that summarizes a set of values.
Types of Averages:
- Mean: The sum of all values divided by the number of values.
  - Formula: (\text{Mean} = \frac{\sum x}{n}) where (n) is the number of values.
- Median: The middle value when data is ordered; if there is an even number of observations, it is the average of the two middle numbers.
- Mode: The value that appears most frequently in a data set.
Applications:
- Used in statistics to summarize data sets.
- Helps in comparison and understanding of distributions.
Properties:
- The mean can be affected by extreme values (outliers).
- The median provides a better measure when the data is skewed.
- The mode is useful for categorical data where we wish to know the most common item.

Algebra focuses on symbols and rules for manipulating those symbols, serving as a foundational area in mathematics.
Expressions consist of numbers, variables, and operators, e.g., (3x + 2), highlighting how values can be combined.
Equations represent the equality of two expressions, as seen in (2x + 3 = 7), crucial for solving unknowns.
Functions define a relationship between inputs and outputs, generally written as (f(x)), emphasizing dependencies between variables.
Linear equations, described by the first degree format (y = mx + b), illustrate straight-line relationships in a coordinate system.
Quadratic equations, characterized by the second degree format (ax^2 + bx + c = 0), represent parabolic curves.
Factoring simplifies expressions, breaking them down into factors; for example, (x^2 - 5x + 6) can be factored into ((x-2)(x-3)).
The quadratic formula, (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}), is essential for finding roots of quadratic equations.
Systems of equations involve multiple equations with several variables, solvable through various methods to find common solutions.
Solving methods include substitution, elimination, and graphing techniques to work with systems effectively.
Inequalities express a range of values, such as (x > 5), indicating possible solutions rather than a singular answer.
Polynomials are algebraic expressions formed of variables with whole number exponents, combined using basic arithmetic operations.

An average provides a measure of central tendency, summarizing a collection of values into a single representative figure.
Types of averages include:
- Mean: Calculated by summing all values and dividing by the total count (Formula: (\text{Mean} = \frac{\sum x}{n})).
- Median: Represents the middle value of an ordered data set; for even counts, the median is the average of the two central numbers.
- Mode: Identifies the most frequently occurring value within a data set, helpful in recognizing common occurrences.
Averages are widely used in statistics for data analysis and to facilitate comparisons across distributions.
The mean can be skewed by outliers, making it less reliable in certain data sets, while the median remains a robust measure in skewed distributions.
The mode is particularly effective for categorical data, helping identify the most prevalent category or item.

Algebra is a mathematical discipline focused on symbols and the manipulation of those symbols to solve equations.
Variables are placeholders, typically represented by letters, that signify numbers or values.
Expressions consist of variables and constants combined through mathematical operations, such as 3x + 5.
Equations assert the equality of two expressions, illustrated by examples like 2x + 3 = 7.
Functions define a relationship between inputs and outputs, often represented as f(x).
Operations in algebra include:
- Addition combines like terms: 2x + 3x simplifies to 5x.
- Subtraction removes terms: 5x - 2x results in 3x.
- Multiplication applies distribution: 2(x + 3) equals 2x + 6.
- Division splits terms: 6x / 2 simplifies to 3x.
Types of equations include:
- Linear equations, expressed as y = mx + b, where m is the slope and b is the y-intercept.
- Quadratic equations in the form ax² + bx + c = 0, solvable by factoring, completing the square, or using the quadratic formula.
- Polynomial equations, involving variables raised to whole number powers.
- Inequalities express ranges of values using symbols such as <, ≤, and ≥.

A percentage represents a fraction of 100, denoted by the symbol %.
Basic calculations involve:
- Finding a percentage of a number using the formula (Percentage / 100) × Total.
- Determining what percentage one number is of another using (Part / Whole) × 100.
When calculating increases or decreases:
- Percentage increase is calculated as ((New Value - Original Value) / Original Value) × 100.
- Percentage decrease uses the formula ((Original Value - New Value) / Original Value) × 100.
Common applications of percentages include financial calculations (taxes, discounts), statistical data representation, and everyday calculations like tips and sales.
Conversion methods involve:
- Changing a fraction to a percentage by multiplying by 100.
- Transforming a decimal to a percentage also by multiplying by 100.
Real-life examples of percentage applications include calculating sales tax or discounts while shopping and analyzing statistical data in reports or surveys.