Algebra Chapter: Definitions and Types of Equations

Algebra

Definitions

• Algebra: a branch of mathematics that deals with variables and their relationships
• Variable: a symbol that represents a value that can change
• Constant: a symbol that represents a fixed value
• Expression: a combination of variables, constants, and mathematical operations
• Equation: a statement that says two expressions are equal

Types of Equations

• Linear equation: an equation in which the highest power of the variable is 1
  • Example: 2x + 3 = 5
• Quadratic equation: an equation in which the highest power of the variable is 2
  • Example: x^2 + 4x + 4 = 0
• Polynomial equation: an equation consisting of variables and coefficients combined using only addition, subtraction, and multiplication
  • Example: x^3 - 2x^2 + x + 1 = 0

Solving Equations

• Linear equations: can be solved using addition, subtraction, multiplication, and division
• Quadratic equations: can be solved using factoring, quadratic formula, or completing the square
• Polynomial equations: can be solved using factoring, remainder theorem, or synthetic division

Arithmetic Progression

Definition

• Arithmetic progression (AP): a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term
• Common difference (d): the fixed constant added to each term to get the next term

Formulae

• nth term of an AP: an = a1 + (n - 1)d
• Sum of first n terms of an AP: Sn = n/2 [2a1 + (n - 1)d]

Properties

• The common difference is constant
• The sequence is infinite
• The terms can be positive, negative, or zero

Examples

• 2, 5, 8, 11, ... is an AP with a1 = 2 and d = 3
• Find the 10th term of the AP: 3, 6, 9, 12, ...

Algebra

Types of Equations

• A linear equation has the highest power of the variable as 1, e.g., 2x + 3 = 5
• A quadratic equation has the highest power of the variable as 2, e.g., x^2 + 4x + 4 = 0
• A polynomial equation consists of variables and coefficients combined using addition, subtraction, and multiplication, e.g., x^3 - 2x^2 + x + 1 = 0

Solving Equations

• Linear equations can be solved using addition, subtraction, multiplication, and division
• Quadratic equations can be solved using factoring, quadratic formula, or completing the square
• Polynomial equations can be solved using factoring, remainder theorem, or synthetic division

Arithmetic Progression

Key Concepts

• An arithmetic progression (AP) is a sequence of numbers where each term is obtained by adding a fixed constant to the previous term
• The fixed constant is called the common difference (d)
• The sequence is infinite, and terms can be positive, negative, or zero

Formulae

• The nth term of an AP is an = a1 + (n - 1)d
• The sum of first n terms of an AP is Sn = n/2 [2a1 + (n - 1)d]

Examples and Applications

• The sequence 2, 5, 8, 11,...is an AP with a1 = 2 and d = 3
• To find the 10th term of the AP: 3, 6, 9, 12,..., use the formula an = a1 + (n - 1)d