Algebra Basics Quiz

Algebra

Definition: Branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations.
Basic Concepts:
- Variables: Symbols (often letters) that represent numbers.
- Constants: Fixed values that do not change.
- Expressions: Combinations of variables and constants using operations (e.g., addition, subtraction).
- Equations: Mathematical statements that two expressions are equal, often containing one or more variables.
Operations:
- Addition and Subtraction: Combining or taking away quantities.
- Multiplication and Division: Scaling quantities or distributing equally.
Linear Equations:
- Form: ( ax + b = c )
- Solutions: Values of ( x ) that make the equation true.
- Graph: Straight line in a coordinate system.
Quadratic Equations:
- Form: ( ax^2 + bx + c = 0 )
- Solutions: Found using factoring, completing the square, or the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
- Graph: Parabola.
Polynomials:
- Definition: Expressions that involve sums of powers of variables (e.g., ( ax^n + bx^{n-1} + ... + c )).
- Degree: Highest power of the variable in the polynomial.
- Operations: Addition, subtraction, multiplication, and division of polynomials.
Factoring:
- Process of breaking down expressions into products of simpler expressions.
- Common methods:
  - Factoring out the greatest common factor (GCF)
  - Using special products (e.g., difference of squares, perfect square trinomials)
  - Trial and error for trinomials.
Functions:
- Definition: A relation that assigns each input exactly one output.
- Notation: ( f(x) ) represents a function of ( x ).
- Types: Linear, quadratic, polynomial, exponential, etc.
Inequalities:
- Expressions that show the relative size or order of two quantities.
- Symbols: ( <, >, \leq, \geq ).
- Solutions: Represented on a number line or graphically.
Systems of Equations:
- Definition: A set of equations with the same variables.
- Methods of solving:
  - Graphical method
  - Substitution method
  - Elimination method
Exponents and Radicals:
- Exponents: Notation representing repeated multiplication (e.g., ( a^n )).
- Laws of Exponents: Rules for simplifying expressions involving exponents.
- Radicals: Roots of numbers (e.g., square root, cube root).
Applications:
- Used in various fields such as physics, engineering, economics, and computer science for problem-solving and modeling real-world scenarios.

Algebra Overview

Branch of mathematics focused on symbols and their manipulation for solving equations.

Basic Concepts

Variables: Represent unknown numbers, usually denoted by letters.
Constants: Fixed numerical values that remain unchanged.
Expressions: Combos of variables and constants using operations like addition and subtraction.
Equations: Statements indicating that two expressions are equal, often incorporating variables.

Operations

Addition and Subtraction: Fundamental operations for combining or removing quantities.
Multiplication and Division: For scaling quantities or equal distribution.

Linear Equations

Standard form: ( ax + b = c ).
Solutions are values of ( x ) making the equation valid.
Graphically represented as a straight line on a coordinate plane.

Quadratic Equations

Standard form: ( ax^2 + bx + c = 0 ).
Solutions obtained via factoring, completing the square, or the quadratic formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
Graphically depicted as a parabola.

Polynomials

Composed of multi-term expressions with variables raised to powers (e.g., ( ax^n + bx^{n-1} + ... + c )).
Degree refers to the highest power of the variable in the polynomial.
Basic operations include addition, subtraction, multiplication, and division.

Factoring

Involves breaking down polynomials into simpler components.
Common techniques:
- Extracting the greatest common factor (GCF).
- Leveraging special products like the difference of squares or perfect square trinomials.
- Employing trial and error for trinomial factoring.

Functions

Defined as relations assigning one unique output for each input.
Notation ( f(x) ) signifies a function of ( x ).
Different types include linear, quadratic, polynomial, and exponential functions.

Inequalities

Indicate the comparative sizes or orders of two values or expressions.
Utilizes symbols such as ( <, \leq, >, \geq ).
Solutions represented visually on a number line or through graphs.

Systems of Equations

Consist of multiple equations sharing at least one variable.
Solved using methods like:
- Graphical representation.
- Substitution for finding exact values.
- Elimination to simplify and solve systems.