Algebra Basics Quiz

Algebra

Definition
- Branch of mathematics dealing with symbols and the rules for manipulating those symbols.
Basic Concepts
- Variables: Symbols (usually letters) representing numbers.
- Constants: Fixed values that do not change.
- Expressions: Combinations of variables, constants, and operations (e.g., 3x + 2).
Operations
- Addition (+): Combining quantities.
- Subtraction (−): Finding the difference between quantities.
- Multiplication (×): Scaling one quantity by another.
- Division (÷): Splitting a quantity into equal parts.
Equations
- Definition: A statement that two expressions are equal (e.g., 2x + 3 = 7).
- Solving Equations: Finding the value of the variable that makes the equation true.
  - Example: To solve 2x + 3 = 7, subtract 3 from both sides, then divide by 2.
Types of Equations
- Linear Equations: Equations of the first degree (e.g., ax + b = 0).
- Quadratic Equations: Second-degree equations (e.g., ax² + bx + c = 0). Can be solved using:
  - Factoring
  - Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
- Polynomial Equations: Involves terms of varying degrees (e.g., ax^n + ... + k).
Functions
- Definition: A relation that assigns exactly one output for each input.
- Types:
  - Linear functions: f(x) = mx + b
  - Quadratic functions: f(x) = ax² + bx + c
  - Polynomial functions: f(x) = a_nx^n + ... + a_1x + a_0
Graphing
- Coordinate Plane: A two-dimensional surface where equations are graphed using an x-axis and y-axis.
- Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
- Intercepts: Points where the graph intersects the axes (x-intercept and y-intercept).
Inequalities
- Definition: Mathematical statements indicating one quantity is larger or smaller than another (e.g., x > 5).
- Solving Inequalities: Similar to solving equations, but remember to reverse the inequality when multiplying or dividing by a negative number.
Factoring
- Definition: Expressing a polynomial as a product of its factors.
- Common methods:
  - Factoring by grouping
  - Using special products (e.g., difference of squares).
Exponents and Radicals
- Exponents: Represents repeated multiplication (e.g., a^n = a × a × ... × a).
- Rules:
  - Product of powers: a^m × a^n = a^(m+n)
  - Power of a power: (a^m)^n = a^(mn)
- Radicals: Inverse operation of exponents (e.g., square root √a).
Systems of Equations
- Definition: A set of two or more equations with the same variables.
- Methods of Solving:
  - Graphical method
  - Substitution method
  - Elimination method

These notes provide a structured overview of key algebra concepts and operations essential for understanding and solving algebraic problems.

Algebra Overview

Branch of mathematics focused on symbols and the manipulation rules of those symbols.

Basic Concepts

Variables: Represent unknown values, typically denoted by letters.
Constants: Unchanging fixed values, such as numbers like 5 or π.
Expressions: Composed of variables, constants, and operations (e.g., 3x + 2).

Operations

Addition (+): Combines quantities into a sum.
Subtraction (−): Determines the difference between two quantities.
Multiplication (×): Increases one quantity by another, often viewed as scaling.
Division (÷): Distributes a quantity into equal parts or groups.

Equations

Definition: A mathematical statement asserting the equality of two expressions (e.g., 2x + 3 = 7).
Solving Equations: Involves finding the variable value that satisfies the equation. For instance, solving 2x + 3 = 7 requires subtracting 3, then dividing by 2.
Types of Equations:
- Linear Equations: First-degree equations (e.g., ax + b = 0).
- Quadratic Equations: Second-degree equations (e.g., ax² + bx + c = 0), solved by factoring or the quadratic formula: x = [−b ± √(b² − 4ac)] / 2a.
- Polynomial Equations: Involve terms of varying degrees (e.g., ax^n + ... + k).

Functions

Definition: Relations that provide one output for each input.
Types:
- Linear Functions: Expressed as f(x) = mx + b.
- Quadratic Functions: Formulated as f(x) = ax² + bx + c.
- Polynomial Functions: General form a_nx^n + ... + a_1x + a_0.

Graphing

Coordinate Plane: A two-dimensional space used for graphing equations, defined by x-axis and y-axis.
Slope-Intercept Form: Represents a line as y = mx + b, where m indicates the slope and b identifies the y-intercept.
Intercepts: Points where the graph crosses the axes, including x-intercept and y-intercept.

Inequalities

Definition: Statements indicating one quantity is larger or smaller than another (e.g., x > 5).
Solving Inequalities: Similar to solving equations, but multiply or divide by a negative number requires reversing the inequality sign.

Factoring

Definition: The process of expressing a polynomial as a product of its factors.
Common Methods:
- Factoring by grouping.
- Utilizing special products, such as the difference of squares.

Exponents and Radicals

Exponents: Indicate repeated multiplication (e.g., a^n = a × a × ... × a).
Rules:
- Product of powers: a^m × a^n = a^(m+n).
- Power of a power: (a^m)^n = a^(mn).
Radicals: Function inversely to exponents, representing roots (e.g., √a).

Systems of Equations

Definition: Consist of two or more equations sharing common variables.
Methods of Solving:
- Graphical method: Visual representation to find intersection.
- Substitution method: Isolating one variable and substituting into another equation.
- Elimination method: Adding or subtracting equations to eliminate a variable.