Math explainers · 25 pages
Algebra explainers
Clear, citable explainers for the foundational concepts of algebra — quadratic equations, slopes, exponents, logarithms, arithmetic sequences, and functions.
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Quadratic equation
A quadratic equation is any mathematical equation that can be rearranged into the standard form ax² + bx + c = 0, where x represents an unknown variable, and a, b, and c are numerical coefficients with a not equal to zero. The term quadratic comes from quadratus, the Latin word for square, reflecting that the variable is squared.
DefinitionQuadratic formula
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It derives directly from completing the square on the standard form of a quadratic equation, ax² + bx + c = 0. By substituting the coefficients a, b, and c into this formula, one can compute the roots of the equation, regardless of whether they are rational, irrational, or complex numbers.
DefinitionSlope-intercept form
Slope-intercept form is represented by the equation y = mx + c (commonly written as y = mx + b in some regions), where m denotes the slope of the line and c represents the y-intercept, which is the point where the line crosses the vertical y-axis (0, c). This form is highly favoured for its simplicity in graphing and analysing linear relationships.
DefinitionSlope formula
The slope formula is m = (y₂ - y₁) / (x₂ - x₁), where m represents the slope, and (x₁, y₁) and (x₂, y₂) are the coordinates of two distinct points on the line. It defines the slope as the ratio of the vertical change (rise) to the horizontal change (run) between these points.
DefinitionParabola
A parabola is a U-shaped curve that represents the geometric graph of a quadratic function. It is defined as the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed straight line (the directrix). The lowest or highest point on the curve is called the vertex.
DefinitionRadicals
Radicals are mathematical expressions that represent the root of a number. The most common is the square root, but they can represent cube roots, fourth roots, or any nth root. A radical consists of three parts: the radical symbol (√), the radicand (the number inside the symbol), and the index (which specifies the degree of the root, omitted for square roots).
DefinitionPolynomials
A polynomial is an algebraic expression featuring one or more terms, where each term consists of a constant coefficient multiplied by variables raised to non-negative integer exponents. The word polynomial is derived from Greek roots meaning many terms, classifying expressions like monomials (one term), binomials (two terms), or trinomials (three terms).
DefinitionInequalities
Inequalities are mathematical relationships comparing expressions using comparison symbols: less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). Unlike equations which yield specific solutions, inequalities generally describe an infinite range or interval of numbers that satisfy the given conditions.
DefinitionCompleting the square
Completing the square is a method used to solve quadratic equations, write quadratic functions in vertex form, or graph circles and parabolas. The process involves taking a quadratic expression x² + bx, finding half of the linear coefficient b, squaring it to get (b/2)², and adding it to create a perfect square trinomial (x + b/2)².
DefinitionDomain and range
Domain is the complete set of all possible independent input values (typically x-values) for which a function is defined and produces a real number. Range is the complete set of all resulting dependent output values (typically y-values) that the function can produce after processing the inputs.
DefinitionExponent rules
Exponent rules (or laws of indices) are mathematical guidelines used to combine and simplify terms with exponents. These include the product rule (xᵃ * xᵇ = xᵃ⁺ᵇ), quotient rule (xᵃ / xᵇ = xᵃ⁻ᵇ), power rule ((xᵃ)ᵇ = xᵃᵇ), negative exponent rule (x⁻ᵃ = 1/xᵃ), and zero exponent rule (x⁰ = 1 for non-zero x).
DefinitionHow to find slope
To find the slope of a straight line, you can use one of three main methods depending on the given information: apply the slope formula m = (y₂ - y₁) / (x₂ - x₁) if you have two points, rearrange the equation into slope-intercept form (y = mx + c) to read the coefficient m, or count the vertical rise over horizontal run directly from a graph.
DefinitionLinear equations
A linear equation is any equation in which the variables are raised only to the first power, never containing exponents, roots, or products of variables. The general form of a linear equation in two variables is ax + by = c, where a, b, and c are constants. Graphing a linear equation always produces a straight line.
DefinitionLogarithm rules
Logarithm rules are mathematical properties that allow for the manipulation of logarithmic terms. The core rules include the product rule (log_b(xy) = log_b(x) + log_b(y)), quotient rule (log_b(x/y) = log_b(x) - log_b(y)), power rule (log_b(x^k) = k * log_b(x)), and the change of base formula (log_b(x) = log_a(x) / log_a(b)).
DefinitionSystems of equations
A system of equations consists of multiple algebraic equations with the same set of variables. The solution is the set of values for the variables that satisfies every equation in the system at the same time. Geometrically, for a system of linear equations, the solution represents the point where the lines intersect.
DefinitionQuadratic function
A quadratic function is a function that can be written in the standard form f(x) = ax² + bx + c, where a, b, and c are real number coefficients and a is not equal to zero. The highest power of the independent variable x is two, which creates a non-linear relationship where outputs change quadratically.
DefinitionLinear function
A linear function is a function of the form f(x) = mx + c, where m represents the constant slope and c represents the y-intercept. It is characterised by a constant rate of change, meaning that for every unit increase in the input x, the output f(x) changes by a fixed amount m.
DefinitionArithmetic sequence
An arithmetic sequence (or arithmetic progression) is a numerical list where each term is found by adding a constant value, known as the common difference (d), to the preceding term. For example, in the sequence 3, 7, 11, 15, the common difference is 4. The nth term is calculated using the formula a_n = a₁ + (n - 1)d.
DefinitionBinomial theorem
The binomial theorem states that any non-negative integer power of a binomial (x + y)ⁿ can be expanded into a sum of terms involving binomial coefficients, calculated as ⁿC_k * x^(n-k) * y^k. The coefficients correspond exactly to the nth row of Pascal’s triangle, simplifying expansions that would otherwise require tedious multiplication.
DefinitionInverse functions
An inverse function, denoted as f⁻¹(x), is a function that undoes the action of the original function f(x). If f maps an input x to an output y, then the inverse f⁻¹ maps y back to x, meaning f⁻¹(f(x)) = x. For a function to have an inverse, it must be a one-to-one function.
DefinitionSlope
Slope (often denoted by the letter m) is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on a line. The formula for the slope of a line passing through points (x₁, y₁) and (x₂, y₂) is m = (y₂ - y₁) / (x₂ - x₁). A slope can be positive, negative, zero, or undefined.
DefinitionFactors
A factor is a number or term that divides a given value completely to yield an integer or polynomial quotient with zero remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. In algebra, factoring (or factorisation) is the process of breaking down polynomials into products of simpler factors, which is essential for solving equations.
DefinitionVectors
A vector is a quantity characterized by both magnitude and direction, distinguishing it from a scalar, which has only magnitude. Vectors are represented geometrically by directed line segments (arrows) and algebraically by ordered lists of components, such as v = [x, y]. They are fundamental in algebra, physics, and computer graphics for representing forces, velocities, and coordinate translations.
DefinitionAlgebra
Algebra is the study of mathematical symbols and the rules for manipulating these symbols. Unlike arithmetic, which deals with concrete numbers, algebra introduces variables (letters like x, y, and z) to stand for unknown values or general relationships. This abstraction allows for the formulation and solution of general mathematical models and equations across science and engineering.
DefinitionFunctions
A function is a mathematical machine that takes an input (x), processes it according to a rule, and returns a single output (y or f(x)). For a relation to be classified as a function, every element in the input set (domain) must map to exactly one element in the output set (codomain or range). Functions are represented using the notation f(x) = y.
