📋 Table of Contents
1. Foundations: Expressions and Equations
Algebra is the branch of mathematics that uses symbols — typically letters — to represent numbers and relationships between them. The two most fundamental building blocks are expressions and equations.
Within an expression, each part separated by a plus or minus sign is a term. In 5x² − 3x + 8, there are three terms. The number multiplying a variable is its coefficient (5 and −3 above). The number with no variable is a constant (8).
Combining like terms: Only add or subtract terms with the exact same variable and exponent. 3x² + 5x² = 8x² ✓ but 3x² + 5x ≠ 8x² ✗ (different powers). Think of it like combining apples with apples, not apples with oranges.
2. Linear Equations and Slope
A linear equation is any equation whose graph is a straight line — the variable has no exponents greater than 1. The most useful form is slope-intercept form:
Calculating Slope
Given any two points (x₁, y₁) and (x₂, y₂) on a line:
Finding Intercepts
The y-intercept: set x = 0 and solve. The x-intercept (where the line crosses the x-axis): set y = 0 and solve. For y = 2x − 6: y-intercept is (0, −6); x-intercept is y=0 → 2x = 6 → x = 3, so (3, 0).
3. Functions and Their Graphs
A function is a rule assigning exactly one output to each input. Written f(x) (read "f of x"), where x is the input and f(x) is the output. The set of valid inputs is the domain; the set of possible outputs is the range.
Vertical Line Test: A graph represents a function if and only if no vertical line intersects it more than once. A circle fails this test (two y-values for most x-values) — it is a relation, not a function.
Common Domain Restrictions
Not all x-values are valid for every function. Restrictions arise from: division by zero (f(x) = 1/x: x ≠ 0) · even roots of negatives (f(x) = √x: x ≥ 0) · logarithms of non-positives (f(x) = log x: x > 0).
Function Notation and Evaluation
To evaluate f(x) = 4x − 1 at x = 3: f(3) = 4(3) − 1 = 11. The notation f(a) means "substitute a for every x." Composite functions: (f ∘ g)(x) = f(g(x)) — apply g first, then f to the result.
4. Polynomials and Factoring
A polynomial is an expression of one or more terms with variables raised to non-negative integer exponents. The degree is the highest exponent: 3x⁴ − 2x + 1 has degree 4.
Polynomial Operations
Addition/subtraction: Combine like terms. (3x² + 2x) + (x² − 5) = 4x² + 2x − 5.
Multiplication: Use the distributive property (FOIL for binomials). (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6.
Factoring Techniques
6x² + 9x = 3x(2x + 3) — factor out the largest common factor first.
a² − b² = (a + b)(a − b). Example: x² − 25 = (x + 5)(x − 5).
x² + 5x + 6 = (x + 2)(x + 3). Find two numbers that multiply to 6 and add to 5.
For 4-term polynomials: 2x³ + 4x² + 3x + 6 → 2x²(x + 2) + 3(x + 2) = (2x² + 3)(x + 2).
5. Quadratic Equations
A quadratic equation has the standard form ax² + bx + c = 0 (a ≠ 0). Its graph is a parabola — U-shaped if a > 0, ∩-shaped if a < 0. The solutions (values of x that satisfy the equation) are called roots or zeros.
The Discriminant
The expression b² − 4ac under the radical is the discriminant. It predicts the number and type of solutions: positive → 2 distinct real roots · zero → 1 repeated real root (vertex on x-axis) · negative → no real roots (2 complex/imaginary roots).
6. Inequalities and Absolute Value
Inequalities compare two quantities using <, >, ≤, or ≥. They are solved like equations with one critical rule: multiplying or dividing both sides by a negative number reverses the inequality sign.
The flip rule: −2x > 8 → divide both sides by −2 → x < −4. The sign flips from > to <. Forgetting this is the #1 inequality mistake.
Absolute value |x| represents distance from zero — always non-negative. |x| = 5 gives two solutions: x = 5 or x = −5. |x| < 5 gives −5 < x < 5 (interval). |x| > 5 gives x < −5 or x > 5 (two separate intervals).
7. Advanced Topics: Radicals, Rationals, Matrices
Radicals and Exponents
A radical (√) denotes a root. Key rules: √(ab) = √a · √b · (√a)² = a · √a² = |a|. Exponent rules: xᵃ · xᵇ = xᵃ⁺ᵇ · (xᵃ)ᵇ = xᵃᵇ · x⁻ᵃ = 1/xᵃ · x^(1/n) = ⁿ√x.
Rational Numbers and Proportions
Rational numbers are expressible as fractions p/q (integers, fractions, repeating decimals). A proportion states two ratios equal: a/b = c/d. Cross-multiply to solve: ad = bc. Proportions model scale, unit conversion, and percent problems.
Matrices
A matrix is a rectangular array of numbers. Two matrices of the same dimensions are added element-by-element. Matrix multiplication follows the rows-times-columns rule (not commutative: AB ≠ BA in general). Matrices are used to solve systems of equations efficiently.
8. Key Terms Glossary
A letter (x, y, n…) representing an unknown or changing quantity.
A statement of equality between two expressions, solvable for an unknown.
Rise over run — the rate of change of a linear function.
Where a graph crosses an axis; y-intercept at x=0, x-intercept at y=0.
The number multiplying a variable in a term (in 7x², the coefficient is 7).
Indicates repeated multiplication; in x⁵, the exponent 5 means x×x×x×x×x.
A relation where each input has exactly one output.
The set of all valid input (x) values for a function.
A degree-2 polynomial equation; graph is a parabola.
A number or expression that divides another evenly; factoring reverses multiplication.
9. Frequently Asked Questions
What is the difference between an expression and an equation?
An expression (3x + 5) has no equals sign — it represents a quantity. An equation (3x + 5 = 14) asserts equality and can be solved. Expressions are simplified; equations are solved. Think of it this way: an expression is like a noun phrase ("the sum of three x and five"); an equation is a complete sentence ("the sum of three x and five equals fourteen").
How do you solve a quadratic equation?
Four methods: (1) Factoring — rewrite as two factors = 0, then set each to zero. (2) Square root method — isolate x² and take the square root (works when b = 0). (3) Completing the square — convert to (x + h)² = k form. (4) Quadratic formula: x = (−b ± √(b²−4ac)) / 2a — always works. Start with factoring; use the formula when factoring isn't obvious.
What is the slope of a line and how is it calculated?
Slope (m) measures steepness and direction: m = rise/run = (y₂ − y₁)/(x₂ − x₁). Positive slope goes up left-to-right (like climbing a hill); negative goes down; zero is horizontal; undefined is vertical. In y = mx + b, m tells you that for every 1 unit right, the line goes m units up (or down if m is negative).
What is a function in algebra?
A function is a rule where every input has exactly one output. Written f(x), it maps each x-value (domain) to one y-value (range). Think of a vending machine: each button press (input) delivers exactly one item (output). The vertical line test confirms a graph is a function if no vertical line crosses it twice. Functions model countless real situations: distance vs. time, cost vs. quantity, temperature vs. elevation.
Why is algebra important in real life?
Algebra builds logical, systematic thinking applied everywhere: personal finance (interest = principal × rate × time → I = Prt), physics (F = ma), computer science (algorithms are algebraic procedures), business (profit = revenue − cost), and cooking (scaling recipes is proportional reasoning). Beyond practical applications, algebra develops the ability to represent unknown situations symbolically and reason toward a solution — a skill fundamental to any complex problem-solving.