Algebra Class: Algebraic Equations

Algebraic Equations

Definition

• An algebraic equation is an equation involving variables and constants, and only uses addition, subtraction, multiplication, and division.
• It is written in the form: ax^n + bx^(n-1) + ... + cx + d = 0, where a, b, ..., c, d are constants and x is the variable.

Types of Algebraic Equations

• Linear Equations: Degree of the equation is 1 (e.g. 2x + 3 = 5)
• Quadratic Equations: Degree of the equation is 2 (e.g. x^2 + 4x + 4 = 0)
• Cubic Equations: Degree of the equation is 3 (e.g. x^3 - 2x^2 - 5x + 1 = 0)
• Polynomial Equations: Degree of the equation is greater than 3 (e.g. x^4 - 3x^3 + 2x^2 - x + 1 = 0)

Solving Algebraic Equations

• Factoring: Expressing the equation as a product of simpler expressions (e.g. x^2 + 5x + 6 = (x + 3)(x + 2) = 0)
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a, used to solve quadratic equations
• Graphical Method: Solving equations by finding the points of intersection between the graph of the equation and the x-axis

Properties of Algebraic Equations

• Addition and Subtraction: Equations can be added or subtracted, as long as they have the same variables and coefficients.
• Multiplication and Division: Equations can be multiplied or divided, as long as they have the same variables and coefficients.
• Distributive Property: a(b + c) = ab + ac, used to expand equations.

Algebraic Equations

• Algebraic equations involve variables and constants, and only use addition, subtraction, multiplication, and division.

Definition and Form

• The general form of an algebraic equation is ax^n + bx^(n-1) +...+ cx + d = 0.
• In this form, a, b,..., c, d are constants, and x is the variable.

Types of Algebraic Equations

• Linear Equations: Have a degree of 1, such as 2x + 3 = 5.
• Quadratic Equations: Have a degree of 2, such as x^2 + 4x + 4 = 0.
• Cubic Equations: Have a degree of 3, such as x^3 - 2x^2 - 5x + 1 = 0.
• Polynomial Equations: Have a degree greater than 3, such as x^4 - 3x^3 + 2x^2 - x + 1 = 0.

Solving Algebraic Equations

• Factoring: Involves expressing the equation as a product of simpler expressions, such as x^2 + 5x + 6 = (x + 3)(x + 2) = 0.
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a, used to solve quadratic equations.
• Graphical Method: Solves equations by finding the points of intersection between the graph of the equation and the x-axis.

Properties of Algebraic Equations

• Addition and Subtraction: Equations can be added or subtracted if they have the same variables and coefficients.
• Multiplication and Division: Equations can be multiplied or divided if they have the same variables and coefficients.
• Distributive Property: a(b + c) = ab + ac, used to expand equations.

Learn about algebraic equations, including linear and quadratic equations, and their definitions and types. Understand the forms of equations and how to work with them.