Algebra

What is Algebra?

Algebra is one of the broadest and most fundamental branches of mathematics. At its core, algebra is about finding unknown values by using letters (called variables) to represent numbers in equations and formulas. The word "algebra" comes from the Arabic word al-jabr, meaning "reunion of broken parts," from the title of a 9th-century book by mathematician al-Khwarizmi.

Algebra provides the language and tools that are essential to nearly every area of mathematics, science, engineering, economics, and computer science. When you learn algebra, you're not just learning to solve equations — you're learning to think logically and abstractly.

Variables and Expressions

A variable is a symbol (usually a letter like x, y, or z) that represents an unknown or changeable value. An algebraic expression is a combination of variables, numbers, and operations.

Key Terminology

• Constant: A fixed value, such as 5, -3, or π
• Variable: A symbol representing an unknown value, like x or y
• Coefficient: The number multiplied by a variable, e.g., in 7x, the coefficient is 7
• Term: A single number, variable, or product of numbers and variables (e.g., 3x², -5y, 12)
• Expression: A combination of terms connected by + or - signs (e.g., 3x² + 2x - 5)

Order of Operations (PEMDAS)

When evaluating expressions, follow this order:

1. Parentheses — evaluate expressions inside parentheses first
2. Exponents — evaluate powers and roots
3. Multiplication and Division — from left to right
4. Addition and Subtraction — from left to right

Linear Equations

A linear equation is an equation where the highest power of the variable is 1. The graph of a linear equation is always a straight line.

Standard Form

Where a, b, and c are constants, and a ≠ 0.

Solving Linear Equations

The goal is to isolate the variable on one side of the equation using inverse operations:

Slope-Intercept Form

Where m is the slope (rate of change) and b is the y-intercept (where the line crosses the y-axis).

• The slope m = (y₂ - y₁) / (x₂ - x₁) measures steepness
• Positive slope: line goes up from left to right
• Negative slope: line goes down from left to right
• Zero slope: horizontal line
• Undefined slope: vertical line

Inequalities

An inequality compares two expressions using symbols like <, >, ≤, or ≥. Unlike equations, inequalities have a range of solutions.

Solving Inequalities

Solve inequalities the same way as equations, with one critical rule:

Compound Inequalities

A compound inequality combines two inequalities joined by "and" or "or":

• "And" inequality: Both conditions must be true (intersection). Example: -3 < x < 5
• "Or" inequality: At least one condition must be true (union). Example: x < -2 or x > 4

Polynomials

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents.

Classification by Degree

• Constant: degree 0 (e.g., 7)
• Linear: degree 1 (e.g., 3x + 2)
• Quadratic: degree 2 (e.g., x² - 4x + 3)
• Cubic: degree 3 (e.g., 2x³ + x² - 5x + 1)
• Quartic: degree 4
• Quintic: degree 5

Polynomial Operations

Addition/Subtraction: Combine like terms (same variable and exponent).

Multiplication: Use the distributive property (FOIL for binomials).

Quadratic Equations

A quadratic equation has the standard form:

Methods for Solving Quadratics

1. Factoring

If the quadratic can be written as a product of two binomials, set each factor equal to zero.

2. Quadratic Formula

This formula works for any quadratic equation. The discriminant Δ = b² - 4ac determines the nature of the roots:

• Δ > 0: Two distinct real roots
• Δ = 0: One repeated real root
• Δ < 0: Two complex conjugate roots

3. Completing the Square

Transform the equation into the form (x + p)² = q, then take the square root of both sides.

The Vertex Form

The vertex of the parabola is at the point (h, k). If a > 0, the parabola opens upward; if a < 0, it opens downward.

Functions

A function is a rule that assigns to each input exactly one output. We write f(x) to denote the output of function f when the input is x.

Function Notation

If f(x) = 2x + 3, then:

• f(0) = 2(0) + 3 = 3
• f(4) = 2(4) + 3 = 11
• f(-1) = 2(-1) + 3 = 1

Domain and Range

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

Types of Functions

• Linear function: f(x) = mx + b (straight line)
• Quadratic function: f(x) = ax² + bx + c (parabola)
• Absolute value function: f(x) = |x| (V-shape)
• Exponential function: f(x) = aˣ (rapid growth or decay)
• Logarithmic function: f(x) = log(x) (inverse of exponential)
• Polynomial function: f(x) = aₙxⁿ + ... + a₁x + a₀
• Rational function: f(x) = p(x)/q(x) where p and q are polynomials

Function Composition

First apply g to x, then apply f to the result.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously.

Methods of Solving

1. Substitution Method

Solve one equation for one variable, then substitute into the other equation.

2. Elimination Method

Add or subtract equations to eliminate one variable.

3. Graphing Method

Graph both equations and find the intersection point(s).

Types of Solutions

• One solution: Lines intersect at exactly one point (consistent and independent)
• No solution: Lines are parallel (inconsistent)
• Infinitely many solutions: Lines are the same (consistent and dependent)

Exponents and Radicals

Exponents represent repeated multiplication. Understanding the laws of exponents is crucial for simplifying expressions.

Laws of Exponents

• Product Rule: aᵐ · aⁿ = aᵐ⁺ⁿ
• Quotient Rule: aᵐ / aⁿ = aᵐ⁻ⁿ
• Power Rule: (aᵐ)ⁿ = aᵐⁿ
• Zero Exponent: a⁰ = 1 (when a ≠ 0)
• Negative Exponent: a⁻ⁿ = 1/aⁿ
• Fractional Exponent: a^(m/n) = ⁿ√(aᵐ)

Radicals

A radical is the inverse operation of an exponent. The most common is the square root:

Simplifying Radicals

Factoring Techniques

Factoring is the process of writing an expression as a product of simpler expressions. It's essential for solving equations and simplifying rational expressions.

Common Factoring Techniques

1. Greatest Common Factor (GCF)

Factor out the largest factor common to all terms:

2. Difference of Squares

3. Perfect Square Trinomials

4. Sum and Difference of Cubes

5. Factoring by Grouping