Introduction to Algebra

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols
• These symbols represent quantities without fixed values, known as variables

Basic Operations

• Addition and subtraction are fundamental operations
• Multiplication and division are also fundamental operations
• These operations are applied to both numbers and variables

Variables and Constants

• Variables are symbols (usually letters) representing unknown or changeable values (e.g., x, y)
• Constants are fixed numerical values (e.g., 2, 7, π)

Expressions

• An algebraic expression is a combination of variables, constants, and operations (e.g., 3x + 5, y^2 - 2x)
• Expressions do not contain an equals sign

Equations

• An algebraic equation is a statement showing the equality of two expressions (e.g., 3x + 5 = 14)
• Equations contain an equals sign, indicating that the expressions on either side have the same value

Solving Equations

• Solving an equation involves finding the value(s) of the variable(s) that make the equation true
• This is typically done by isolating the variable on one side of the equation using inverse operations

Linear Equations

• A linear equation is an equation where the highest power of the variable is 1 (e.g., 2x + 3 = 7)
• Linear equations can be represented graphically as a straight line

Solving Linear Equations

• Use inverse operations to isolate the variable
• Example: Solve 2x + 3 = 7
  • Subtract 3 from both sides: 2x = 4
  • Divide both sides by 2: x = 2
  • Solution: x = 2

Quadratic Equations

• A quadratic equation is an equation where the highest power of the variable is 2 (e.g., ax^2 + bx + c = 0)
• Quadratic equations have a general form of ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0

Solving Quadratic Equations

• Factoring: Express the quadratic expression as a product of two linear factors
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a)
• Completing the Square: Manipulate the equation to form a perfect square trinomial

Factoring

• Factoring involves expressing an algebraic expression as a product of its factors
• Example: Factor x^2 + 5x + 6
  • Find two numbers that multiply to 6 and add to 5 (2 and 3)
  • (x + 2)(x + 3)
  • Factored form: (x + 2)(x + 3)

Quadratic Formula

• Used to find the solutions (roots) of the quadratic equation ax^2 + bx + c = 0
• Formula: x = (-b ± √(b^2 - 4ac)) / (2a)
• The discriminant (b^2 - 4ac) determines the nature of the roots

Systems of Equations

• A set of two or more equations containing the same variables
• Solving a system of equations involves finding values for the variables that satisfy all equations simultaneously

Methods for Solving Systems of Equations

• Substitution: Solve one equation for one variable and substitute that expression into the other equation
• Elimination (Addition/Subtraction): Add or subtract the equations to eliminate one variable

Inequalities

• An inequality is a statement that compares two expressions using inequality symbols
• Inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to)

Solving Inequalities

• Similar to solving equations, but multiplying or dividing by a negative number reverses the inequality sign
• Example: Solve -2x < 6
  • Divide both sides by -2 (and reverse the inequality sign): x > -3
  • Solution: x > -3

Polynomials

• An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and non-negative integer exponents
• General form: anx^n + an-1x^(n-1) + ... + a1*x + a0, where a's are coefficients and n is a non-negative integer

Operations with Polynomials

• Addition: Combine like terms (terms with the same variable and exponent)
• Subtraction: Distribute the negative sign and combine like terms
• Multiplication: Use the distributive property to multiply each term in one polynomial by each term in the other polynomial

Exponents

• An exponent indicates how many times a base number is multiplied by itself
• Example: x^3 (x is the base, 3 is the exponent)

Rules of Exponents

• Product of Powers: a^m * a^n = a^(m+n)
• Quotient of Powers: a^m / a^n = a^(m-n)
• Power of a Power: (a^m)^n = a^(m*n)
• Zero Exponent: a^0 = 1 (if a ≠ 0)
• Negative Exponent: a^(-n) = 1 / a^n

Radicals

• A radical (√) indicates the root of a number
• Square root (√x) is the number that, when multiplied by itself, equals x
• Cube root (∛x) is the number that, when multiplied by itself twice, equals x

Simplifying Radicals

• Factor the radicand (the number under the radical) into perfect square factors
• Example: √20 = √(4 * 5) = √4 * √5 = 2√5

Logarithms

• The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number
• Notation: log_b(x) = y means b^y = x

Properties of Logarithms

• Product Rule: log_b(mn) = log_b(m) + log_b(n)
• Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
• Power Rule: log_b(m^p) = p * log_b(m)
• Change of Base Formula: log_b(a) = log_c(a) / log_c(b)

Functions

• A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output
• Notation: f(x) represents the output of the function f when the input is x

Types of Functions

• Linear Functions: f(x) = mx + b (straight line)
• Quadratic Functions: f(x) = ax^2 + bx + c (parabola)
• Exponential Functions: f(x) = a^x
• Logarithmic Functions: f(x) = log_b(x)

Graphing

• Graphing involves plotting points on a coordinate plane to represent equations or functions
• Coordinate plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical)

Graphing Linear Equations

• Find two points that satisfy the equation and draw a line through them
• Use the slope-intercept form (y = mx + b) to identify the slope (m) and y-intercept (b)

Graphing Quadratic Equations

• The graph is a parabola
• Find the vertex (the maximum or minimum point) and plot additional points

Domain and Range

• Domain: The set of all possible input values (x-values) for a function
• Range: The set of all possible output values (y-values) for a function