Algebra: Types of Algebraic Equations

Study Notes

An algebraic equation is an equation that involves variables and constants, and uses only addition, subtraction, multiplication, and division, as well as raising to a power.
The variables are typically represented by letters, and the constants are numbers.

Linear Equations: Equations in which the highest power of the variable(s) is 1.
- Example: 2x + 3 = 5
Quadratic Equations: Equations in which the highest power of the variable(s) is 2.
- Example: x^2 + 4x + 4 = 0
Polynomial Equations: Equations in which the variables are raised to non-negative integer powers.
- Example: x^3 + 2x^2 - 7x - 12 = 0
Rational Equations: Equations in which the variables are expressed as the ratio of two polynomials.
- Example: (x + 1) / (x - 1) = 2

Addition and Subtraction: Isolate the variable by adding or subtracting the same value to both sides of the equation.
Multiplication and Division: Isolate the variable by multiplying or dividing both sides of the equation by the same non-zero value.
Factoring: Express the equation as a product of simpler expressions, and then solve for the variable.
Quadratic Formula: A formula used to solve quadratic equations, given by x = (-b ± √(b^2 - 4ac)) / 2a.

Problem-Solving: Algebraic equations are used to model and solve real-world problems, such as finding the cost of goods, the area of a rectangle, or the speed of an object.
Science and Engineering: Algebraic equations are used to describe the laws of physics, chemistry, and other scientific fields, and to design and optimize systems.
Computer Science: Algebraic equations are used in computer programming and coding, and are essential for developing algorithms and solving computational problems.

An algebraic equation involves variables and constants, using operations like addition, subtraction, multiplication, division, and exponentiation.
Variables are represented by letters, while constants are numerical values.

Linear Equations: Highest power of variable is 1.
- Example: 2x + 3 = 5
Quadratic Equations: Highest power of variable is 2.
- Example: x² + 4x + 4 = 0
Polynomial Equations: Variables are raised to non-negative integer powers.
- Example: x³ + 2x² - 7x - 12 = 0
Rational Equations: Variables appear as ratios of two polynomials.
- Example: (x + 1) / (x - 1) = 2

Addition and Subtraction: Isolate the variable by performing the same operation on both sides.
Multiplication and Division: Isolate the variable using multiplication or division by a non-zero value on both sides.
Factoring: Rewrite the equation as a product of simpler expressions for solving.
Quadratic Formula: Used specifically for quadratic equations, expressed as x = (-b ± √(b² - 4ac)) / 2a.

Problem-Solving: Models and solves practical issues, such as costs, areas, or speeds.
Science and Engineering: Describes scientific laws and helps in designing and optimizing systems.
Computer Science: Fundamental in programming, algorithm development, and computational problem-solving.