Algebra

Standard Form of Quadratic Equation — ax² + bx + c = 0

The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$. This article walks through how to convert any quadratic to standard form, three worked examples at increasing difficulty, the most common conversion slips, and how standard form compares to vertex and factored form.

Inequalities - Symbols, Solving, and Graphing Guide

An inequality is a statement that compares two expressions with $<$, $>$, $\leq$, or $\geq$, and unlike an equation it has a range of solutions, not one. This article covers the four symbols and types, how to solve and graph inequalities on a number line, the flip-the-sign-on-negatives rule, interval notation, and the mistakes that lose marks.

Exponents — Definition, Laws, and Examples

An exponent tells you how many times to multiply a number, the base, by itself — so $2^5$ means $2 \times 2 \times 2 \times 2 \times 2 = 32$. This article covers what an exponent is, the parts of a power, the seven laws of exponents at a glance, negative, fractional, zero, and decimal exponents, where exponents show up in the real world, and the slips that cost marks.

Covariance Matrix — Formula, Properties, Examples

A covariance matrix is a square matrix that holds the variance of each variable on its diagonal and the covariance between each pair of variables off the diagonal — and it is always symmetric and positive semi-definite. This article covers the definition, the variance and covariance formulas, the key properties, a full 2×2 worked example, and how the matrix powers principal component analysis.

Cardinality of a Set — Definition, Symbol, Examples

The cardinality of a set is the number of elements it contains, written $|A|$ or $n(A)$ — so the set ${2, 4, 6, 8}$ has cardinality $4$. This article covers the notation, finite versus infinite sets, countable versus uncountable infinity (aleph-null), the power-set rule $|P(A)| = 2^n$, the union formula, and six worked examples.

Homogeneous System of Linear Equations — Guide

A homogeneous system of linear equations is a system where every constant term is zero, written compactly as $AX = 0$. This article covers why it always has the all-zeros (trivial) solution, the exact condition for non-trivial solutions ($\det A = 0$, or more variables than equations), how to solve one step by step, and six worked examples.

Perfect Square Trinomial — Formula & Examples

A perfect square trinomial is a three-term expression that factors into the square of a binomial — $a^2 + 2ab + b^2 = (a+b)^2$ or $a^2 - 2ab + b^2 = (a-b)^2$. This article shows you the formula, how to recognise one on sight, how to factor it, the most common sign and middle-term mistakes, and six worked examples.

Square Root of 65 — Value, Method & Examples

The square root of 65 is approximately $\sqrt{65} \approx 8.0623$ — irrational, non-terminating, and already in simplest radical form because $65$ has no square factor. This article gives the value in exact and decimal form, three ways to compute it, where $\sqrt{65}$ shows up, and the slips students make most often.

Square Root of 196 — Value, Method & Examples

The square root of 196 is exactly $14$, because $14 \times 14 = 196$ — so $196$ is a perfect square with a whole-number root. This article gives the value, two reliable methods to find it, where $\sqrt{196}$ appears, and the mistakes that trip students up.

Square Root of 21 — Value, How to Find It, and Examples

The square root of 21 is about $4.583$, and because $21 = 3 \times 7$ has no repeated prime factor, $\sqrt{21}$ is irrational and already in simplest radical form. This article gives the exact and decimal value, two by-hand methods, where $\sqrt{21}$ shows up, and the mistakes students make most.

Square Root of 14 — Value, How to Find It, and Examples

The square root of 14 is about $3.742$, and because $14 = 2 \times 7$ has no repeated prime factor, $\sqrt{14}$ is irrational and already in simplest radical form. This article gives the exact and decimal value, two by-hand methods, where $\sqrt{14}$ shows up, and the slips students make most.

Square Root of 100 — Value, Method & Examples

The square root of 100 is exactly $10$, because $10 \times 10 = 100$ — so $100$ is a perfect square with a clean whole-number root. This article gives the value, three ways to compute it, where $\sqrt{100}$ shows up, and the slip-ups students make most often.

Square Root of 25 — Value, Method, and Examples

The square root of 25 is exactly $5$, because $5 \times 5 = 25$ — making $25$ a perfect square with a whole-number root. This article gives the value, three ways to find it, where $\sqrt{25}$ turns up, and why the radical answer is $5$ rather than $\pm 5$.

Square Root of 8 — Value, Simplified Form, and Examples

The square root of 8 is $2\sqrt{2}$, approximately $2.828$, and it is irrational. Unlike a prime radicand, $\sqrt{8}$ simplifies because $8 = 4 \times 2$. This article gives the exact and decimal value, two by-hand methods, where $2\sqrt{2}$ shows up, and the mistakes students make most.

Square Root of 5 — Value, How to Find It, and Examples

The square root of 5 is about $2.236$, and because $5$ is prime, $\sqrt{5}$ is irrational and already in simplest radical form. This article gives the exact and decimal value, two by-hand methods, the link between $\sqrt{5}$ and the golden ratio, and the slips students make most.

Square 1 to 30 — Table of Square Values and Patterns

The squares from 1 to 30 run from $1^2 = 1$ to $30^2 = 900$. This article gives the complete table, the odd-number pattern that links each square to the next, the fastest memory tricks, and what makes each of these values a perfect square.

Square Root of 3 — Value, How to Find It, and Examples

The square root of 3 is about $1.732$, and because $3$ is prime, $\sqrt{3}$ is irrational and already in simplest radical form. This article gives the exact and decimal value, two ways to compute it by hand, where $\sqrt{3}$ turns up in geometry, and the mistakes students make most.

Prove That Root 3 Is Irrational — Full Proof

To prove that root 3 is irrational, assume the opposite — that $\sqrt{3} = \tfrac{p}{q}$ for coprime integers — then show 3 must divide both $p$ and $q$, contradicting "coprime." This article gives the full proof by contradiction step by step, the key divisibility lemma it rests on, a long-division check, worked examples, and the mistakes that quietly break the argument.

Log Base 2 — Binary Logarithm, Values & Examples

Log base 2 of a number is the power to which 2 must be raised to produce that number — so $\log_2 8 = 3$ because $2^3 = 8$. This article covers what log base 2 (the binary logarithm) means, how to compute it by hand and with the change-of-base formula, a value table, its properties, where it powers computer science, and the slips to avoid.

Radicand — Definition, Symbol, and Examples

The radicand is the number or expression sitting under a radical (root) sign — in $\sqrt{49}$, the radicand is 49. This article covers the radicand's definition, the three parts of a radical, how the radicand differs from the radical, what kinds of values a radicand can hold, and the slips students make.

Rational Root Theorem - Steps, Proof, and Examples

The rational root theorem says that any rational root of a polynomial with integer coefficients has the form $\pm\tfrac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient. This article shows how to list every candidate, test them efficiently, the proof, six worked examples, and the theorem's honest limits.

Simplifying Rational Expressions - Steps and Examples

Simplifying rational expressions means factoring the numerator and denominator, canceling the factors they share, and stating the restrictions that keep the denominator from being zero. This article gives the three-step method, six worked examples, the cancel-only-factors rule, and why the restrictions must come from the original expression.

Literal Equations - Definition, Steps, and Examples

A literal equation is an equation with two or more variables — usually a formula — that you rearrange to isolate one chosen variable. This article shows the inverse-operation method, six worked examples (from $A = lw$ to multi-fraction formulas), how to factor when the target appears twice, restrictions, and the mistakes that flip a formula.

Two-Step Equations - Steps, Rules, and Examples

A two-step equation is a linear equation of the form $ax + b = c$ that takes exactly two inverse operations to solve: undo the addition or subtraction first, then undo the multiplication or division. This article gives the method, six worked examples (integers, fractions, decimals, negatives, and a word problem), the order rule, and the mistakes that cost marks.

Expression, Term, Factor, Coefficient — Definitions, Examples

Every algebraic expression is built from terms (parts joined by + or −), factors (parts multiplied inside a term), and coefficients (the number multiplying the variable). This article defines all four, shows exactly how to spot each one, covers like versus unlike terms, and works through examples that take an expression apart piece by piece.

Difference Between Permutation and Combination

Permutation vs Combination is whether order matters: a permutation counts arrangements (order matters), while a combination counts selections (order doesn't). This article covers both definitions, the $nPr$ and $nCr$ formulas derived from scratch, the relationship between them, and how to tell which one a problem needs.

Standard Notation — Standard Form of Numbers, Examples

Standard notation is the ordinary way of writing a number with its digits in their place-value positions — like $4{,}500{,}000$ rather than $4.5 \times 10^6$. This article covers what standard notation means, how it differs from scientific and expanded form, converting between them, and writing very large and very small numbers.

Roster Notation — Roster Form of a Set With Examples

Roster notation writes a set by listing all its elements inside curly braces separated by commas, like ${1, 2, 3}$. This article covers the rules for writing roster form, finite and infinite sets, the contrast with set-builder notation, worked examples, and exactly where roster form stops being practical.

Venn Diagram — Union, Intersection, Complement Examples

A Venn diagram is a picture of overlapping circles that shows how sets share or separate their elements. This article covers the union, intersection, and complement operations, two-set and three-set diagrams, every region they create, the counting formulas, and step-by-step word problems.

Sum of Cubes of n Natural Numbers — Formula & Proof

The sum of cubes of n natural numbers is $1^3 + 2^3 + \cdots + n^3 = \left[\dfrac{n(n+1)}{2}\right]^2$ — which is exactly the square of the sum of the first $n$ numbers. This article covers the formula, two proofs (telescoping and induction), the even and odd variants, and six worked examples.

Fibonacci Numbers — Golden Ratio and Binet's Formula

The Fibonacci numbers are the sequence $0, 1, 1, 2, 3, 5, 8, 13, \ldots$, where each term is the sum of the two before it: $F_n = F_{n-1} + F_{n-2}$. This article covers the recurrence, the full list, the golden-ratio connection, Binet's formula for the $n$th term, key properties, and six worked examples.

Vector Equation — Line and Plane Forms, Examples

A vector equation describes a line or a plane using position vectors and a parameter, most often the line form $\vec{r} = \vec{a} + \lambda\vec{b}$ — start at point $\vec{a}$, then slide along direction $\vec{b}$. This article covers the line form, the two-point form, the plane forms, what the parameter $\lambda$ does, how to convert to Cartesian form, and six worked examples.

Product of Vectors — Dot and Cross Product Guide

The product of vectors comes in two distinct forms: the dot product $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$, which returns a number, and the cross product $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta , \hat{n}$, which returns a vector. This article maps both, shows when each one is the right tool, and extends the idea to the scalar triple product.

Increasing and Decreasing Intervals — Method, Examples

A function's increasing and decreasing intervals are the stretches of input where its output is climbing or falling — found by checking where the first derivative $f'(x)$ is positive or negative. This article covers the definition, the sign-chart method using critical points, the graph approach, the open-versus-closed interval question, and six worked examples.

Surjective Function — Onto Function, Proof, Examples

A surjective function, also called an onto function, is one where every element of the codomain is hit by at least one input — nothing in the target set is left out. This article covers the formal definition, how to prove a function is surjective, the difference from injective and bijective functions, the codomain-versus-range distinction, and six worked examples.

Greatest Integer Function — Graph, Domain, Examples

The greatest integer function, written $\lfloor x \rfloor$, returns the largest integer that is less than or equal to $x$ — so it rounds every number down to a whole number. This article covers its definition, the staircase graph, domain and range, key properties, the rule for negative numbers, and six worked examples.

Relations and Functions — Difference, Types, Examples

A relation is any set of input-output pairs, while a function is the special kind of relation where every input has exactly one output. This article covers the difference between the two, how to read domain and range, the vertical line test that separates them, the main types of each, and six worked examples.

One to One Function — Definition, Graph, Examples

A one to one function is a function where no two different inputs ever share the same output, so distinct $x$-values always give distinct $y$-values. This article covers the injective definition, the horizontal line test, how to prove a function is one to one algebraically, why only these functions have inverses, and six worked examples.

Symmetric Matrix — Properties & Examples

A symmetric matrix is a square matrix that equals its own transpose, $A^T = A$, so the entry in row $i$, column $j$ always matches the entry in row $j$, column $i$. This article covers the definition, the key properties (real eigenvalues, orthogonal eigenvectors, symmetric inverse), the symmetric-versus-skew-symmetric distinction, the decomposition theorem, and six worked examples.

Triangular Matrix — Upper, Lower, Properties, and Examples

A triangular matrix is a square matrix in which all the entries on one side of the main diagonal are zero — upper triangular when the zeros sit below the diagonal, lower triangular when they sit above. This article covers both types, the strictly-triangular and unit-triangular variants, the key properties, the determinant shortcut, and six worked examples.

Cofactor Matrix — Formula, Signs, Examples

The cofactor matrix of a square matrix is built by replacing each entry with its cofactor — the determinant of the smaller matrix left after deleting that entry's row and column, signed by $(-1)^{i+j}$. This article covers minors versus cofactors, the checkerboard sign pattern, the formula, how the cofactor matrix builds the adjugate, determinant, and inverse, and six worked examples.

Invertible Matrix — Theorem, Properties, Examples

An invertible matrix is a square matrix $A$ for which a matrix $A^{-1}$ exists with $AA^{-1} = A^{-1}A = I$ — equivalently, a square matrix whose determinant is non-zero. This article covers the definition, the invertible matrix theorem, the key properties, how to check invertibility, and six worked examples.

Inverse of a Matrix — Definition & Formula

The inverse of a matrix $A$ is the square matrix $A^{-1}$ satisfying $AA^{-1} = A^{-1}A = I$, and it exists only when $A$ is square with a non-zero determinant. This article covers what an inverse really is, the fast 2x2 formula, the adjugate-over-determinant method for any size, the invertibility condition, and six worked examples.

Inverse of a 3x3 Matrix — Formula & Examples

The inverse of a 3x3 matrix $A$ is the matrix $A^{-1}$ with $AA^{-1} = A^{-1}A = I$, found by $A^{-1} = \frac{1}{\det A},\text{adj}(A)$ whenever $\det A \neq 0$. This article covers the 3x3 determinant, the cofactor and adjugate steps, the row-reduction alternative, six worked examples, and the mistakes that quietly break the calculation.

Square Root of 16 — Value, Methods, Why It Is 4

The square root of 16 is exactly $4$ — rational, an integer, no decimal tail. The principal square root is $4$; both $4$ and $-4$ satisfy $x^2 = 16$. This article covers why $\sqrt{16} = 4$, three methods (prime factorization, repeated subtraction, long division), where $\sqrt{16}$ shows up, and the slips students make on perfect-square radicals.

Square Root of 4 — Why It Equals 2, Methods, Examples

The square root of 4 is exactly $2$ — rational, an integer, no decimal tail. The principal (positive) square root is $2$, though $-2$ also squares to $4$. This article covers why $\sqrt{4} = 2$, three methods to verify it (prime factorization, repeated subtraction, long division), where $\sqrt{4}$ shows up, and the slip that trips students at the boundary between $\sqrt{4}$ and $\pm 2$.

Square Root of 20 — Simplify, Methods, Value

The square root of 20 is $2\sqrt{5}$ in simplified radical form and approximately $4.4721$ as a decimal — irrational, because the prime $5$ inside the radical has no partner. This article walks through what $\sqrt{20}$ equals, three ways to compute it (prime factorization, repeated subtraction, long division), where it shows up in geometry, and the slips that cost the most marks.

Square Root of 50 — Value, Radical Form, Methods

The square root of 50 is $5\sqrt{2}$ in simplified radical form and approximately $7.0711$ as a decimal — irrational, because the prime $2$ inside the radical has no partner. This article covers the exact value, three methods to compute it (prime factorization, repeated subtraction, long division), where $\sqrt{50}$ shows up in geometry, and the slips students make most often.

Square 1 to 50 — List of Perfect Square Values

The squares from 1 to 50 range from $1^2 = 1$ to $50^2 = 2500$. This article gives the complete list, the patterns that make memorisation faster (the last-digit pattern, the squares-near-50 trick), and the most common contexts where these values appear in school maths.

Like and Unlike Algebraic Terms — Definition, Examples

Like terms in algebra share the same variables raised to the same powers — only their coefficients can differ. Unlike terms differ in their variable parts or exponents. This article covers the definitions, the addition/subtraction rule that applies only to like terms, three worked examples, and the most common slip students make in simplification.

Common Difference — Arithmetic Progression Definition

The common difference $d$ in an arithmetic progression is the fixed value added to each term to get the next. $d = a_n - a_{n-1}$, the same for every consecutive pair. This article covers the definition, three worked examples, the $n$th-term and sum formulas that use $d$, and how to find $d$ when only two non-adjacent terms are known.

Perfect Cube — Definition, List, Examples

A perfect cube is an integer of the form $n^3$ where $n$ is an integer — like $1, 8, 27, 64, 125$. This article covers the definition, the list of perfect cubes from 1 to 1000, three worked examples of identifying perfect cubes, the digital-root test, and how perfect cubes connect to volumes and to factoring sums and differences of cubes.

Logarithms — Definition, Rules, Examples

A logarithm $\log_b(x)$ is the exponent to which the base $b$ must be raised to produce $x$. This article covers the definition as the inverse of exponentiation, the seven log rules (product, quotient, power, change-of-base, identity, zero, base-of-base), three worked examples, and the historical breakthrough Napier made in 1614.

Eigenvectors — Definition, Calculation, Examples

An eigenvector of a square matrix $A$ is a nonzero vector $v$ that gets stretched (not rotated) by the transformation $A$ — its direction is preserved, only its length is scaled. The scaling factor $\lambda$ is the eigenvalue. This article covers the definition, the two-step calculation, three worked examples, and the appearance of eigenvectors in PageRank, PCA, and quantum mechanics.

Absolute Value Function — Graph, Properties, Examples

The absolute value function $f(x) = |x|$ outputs the distance of $x$ from zero — always non-negative. Its graph is a V centred at the origin. This article covers the piecewise definition, the V-shape, transformations $f(x) = a|x - h| + k$, three worked examples, and the most common slip in solving absolute-value equations.

Geometric Sequence — Formula, Sum, Examples

A geometric sequence is a list of numbers where every term after the first is found by multiplying the previous term by a fixed common ratio $r$. This article covers the $n$th-term and sum formulas, three worked examples at Quick/Standard/Stretch tiers, the role of $|r| < 1$ for convergent infinite series, and the difference between a sequence and a series.

Subsets — Definition, Types, Examples

A subset of a set $A$ is any set whose elements are all also elements of $A$ — written $B \subseteq A$. This article covers the definition, the proper/improper distinction, the formula $2^n$ for the total number of subsets of an $n$-element set, three worked examples, and the empty set's role as a subset of every set.

Even and Odd Functions — Definition, Tests, Examples

A function $f$ is even when $f(-x) = f(x)$ (its graph is symmetric about the $y$-axis) and odd when $f(-x) = -f(x)$ (its graph is symmetric about the origin). This article covers the algebraic test, three worked examples at Quick/Standard/Stretch tiers, the addition and multiplication rules, and what happens when a function is neither.

Singular Matrix — Definition, Properties, Examples

A singular matrix is a square matrix whose determinant equals zero. This means it has no inverse, its rows (or columns) are linearly dependent, and the system $Ax = b$ has either no solution or infinitely many. This article covers the definition, the seven equivalent characterisations, three worked examples, and what singularity tells you about a linear system.

Simplifying Exponents — Rules, Laws, Examples

Simplifying exponents means reducing an exponential expression to its shortest equivalent form by applying the seven laws of exponents — product, quotient, power-of-a-power, zero, negative, fractional, and quotient-power. This article covers each rule with examples, three worked problems at Quick/Standard/Stretch tiers, and the most common slips students make.

Binomial — Definition, Examples, Operations

A binomial is an algebraic expression with exactly two unlike terms — like $3x + 5$ or $x^2 - 9$. This article covers the definition, three worked examples at Quick/Standard/Stretch tiers, the four basic operations on binomials (with the FOIL method for multiplication), and the link to the binomial theorem for higher powers.

Symmetric Property — Definition, Examples

The symmetric property of equality says that if one quantity equals another, then the second equals the first — if $a = b$, then $b = a$. This article covers the formal definition, the congruence form, three worked examples at Quick/Standard/Stretch tiers, and where the symmetric property breaks (notably for inequalities).

Orthogonal Matrix — Properties, Examples

An orthogonal matrix is a square matrix $A$ whose transpose equals its inverse — equivalently, $A^T A = AA^T = I$. This article covers the formal definition, the property list, three worked examples at Quick/Standard/Stretch tiers, and the role of orthogonal matrices in 3D rotations, computer graphics, and quantum mechanics.

Monomial — Definition, Examples & Degree

A monomial is an algebraic expression made of a single term — a constant coefficient multiplied by one or more variables, each raised to a non-negative integer exponent. This article covers the definition, the parts of a monomial, how to find the degree, three worked examples, and the line that separates a monomial from a binomial, trinomial, and general polynomial.

Factoring Trinomials — Methods, Examples

Factoring trinomials rewrites $ax^2 + bx + c$ as a product of two binomials by finding two numbers whose product is $ac$ and whose sum is $b$. This article covers the three standard methods (sum-product when $a = 1$, AC/grouping when $a \neq 1$, and the perfect-square shortcut) with three worked examples and the most common slip students make.

Geometric Progressions — Formula, Sum, Examples

A geometric progression is a sequence in which every term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio. This article covers the $n$th-term and sum formulas, three worked examples at Quick/Standard/Stretch tiers, the infinite-sum case, and the 1859 Australian rabbit explosion that turned a geometric progression into a national crisis.

Binomial Theorem — Formula, Expansion, Examples

The binomial theorem gives a closed-form expansion of $(a + b)^n$ as a sum of $n+1$ terms where the coefficients are the binomial numbers $\binom{n}{k}$ — the same numbers in Pascal's triangle. This article covers the formula, the general term, three worked examples at Quick/Standard/Stretch tiers, and the historical thread from Pingala's chandas to Newton's generalisation.

Completing the Square — Method, Formula, Examples

Completing the square rewrites a quadratic $ax^2 + bx + c$ in the form $a(x + h)^2 + k$ by adding and subtracting $(b/2a)^2$. This article covers the four-step method, three worked examples at Quick/Standard/Stretch tiers, the geometric meaning, and how completing the square produces the quadratic formula itself.

Simplifying Expressions — Rules, Methods, Examples

Simplifying expressions means rewriting an algebraic expression in the shortest equivalent form without changing its value, using like-term combination, the distributive property, and the order of operations. This article covers the formal rules, three worked examples at Quick/Standard/Stretch tiers, where simplification shows up in higher math, and the most common slip students make.

Multiplying and Dividing Exponents - Rules, Examples

Multiplying exponents with the same base adds them: $x^m \cdot x^n = x^{m + n}$. Dividing subtracts them: $\tfrac{x^m}{x^n} = x^{m - n}$. The two rules require the same base on both sides — without that, no exponent rule applies. This article covers the product rule, the quotient rule, negative and fractional exponents, the three errors that cost marks, and worked examples.

Explicit Formulas for Sequences - Methods, Examples

An explicit formula for a sequence gives the $n$-th term $a_n$ directly as a function of $n$ — no need to compute the previous terms first. For an arithmetic sequence, the formula is $a_n = a_1 + (n-1)d$. For geometric, $a_n = a_1 \cdot r^{n-1}$. This article covers the derivation, the four most-tested sequence types, worked examples, the three off-by-one mistakes that cost marks, and the mathematicians who built the formulas.

Solve Matrices - Gaussian Elimination, Inverse, Cramer's Rule

Squares and Square Roots - Rules, Properties, Examples

Squares multiply a number by itself; square roots reverse the operation. Learn the rules, perfect squares, irrationality, simplification, and the most common mistakes.

Zeros of a Function - Definition, Methods, Examples

A zero of a function $f$ is any input value $x$ for which $f(x) = 0$. On a graph, the zeros are the $x$-coordinates where the curve crosses or touches the $x$-axis. This article covers the formal definition, the four standard methods for finding zeros multiplicity of zeros, the three common errors, and the historical story that connects zeros to the rise of polynomial equations.

Adding Exponents - Rules, Methods, Examples

Adding exponents follows two rules — only like terms combine, and exponents themselves don't add. Worked examples for same base, different bases, and fractional powers.

Standard to Vertex Form - Conversion, Formula, Examples

Converting from standard form to vertex form turns $ax^2 + bx + c$ into $a(x - h)^2 + k$, where $(h, k)$ is the parabola's vertex. The two reliable methods are completing the square (always works, builds intuition) and the shortcut formula $h = -\tfrac{b}{2a}$, $k = c - \tfrac{b^2}{4a}$. This article covers both methods, three worked examples, the three mistakes that cost marks, the real-world WHY behind the vertex.

Relations in Math - Definition, Types, Examples

A relation in math is a rule that links each element of one set to one or more elements of another set, written as a collection of ordered pairs. This article covers the formal definition, domain and range, the eight standard types (empty, universal, identity, reflexive, symmetric, transitive, equivalence, inverse), how a relation differs from a function, and the three mistakes that cost students marks every term.

Types of Polynomials — Classification with Examples

Polynomials are classified in two independent ways: by degree (constant, linear, quadratic, cubic, quartic, quintic, and higher) and by number of terms (monomial, binomial, trinomial, and beyond). This article walks through every type, gives three worked examples spanning Quick to Stretch, lists the mistakes that cost marks, and credits the people who built the classification.

Factoring Cubic Polynomials — Methods & Examples

To factor a cubic polynomial, three reliable methods cover almost every case — the Rational Root Theorem, the grouping method, and the sum/difference of cubes identities. This article walks through all three with three worked examples spanning Quick to Stretch, the mistakes that cost marks, and the people who built the framework.

Properties of Equality — List, Definitions, Examples

The properties of equality are the nine algebraic rules that say if you do the same thing to both sides of an equation, the equation stays true. This article lists all nine, gives three worked examples from Quick to Stretch, walks through the mistakes that cost marks, and credits the mathematicians who codified the axioms.

Standard Form of a Polynomial — Rules and Examples

The standard form of a polynomial writes terms in descending order of degree, with no like terms and no zero-coefficient terms. This article covers the two rules of standard form, three worked examples, the common slips, and a side-by-side comparison of polynomial types — monomial through quintic — organised by degree and number of terms.

Rational Exponents — Rules, Properties, Examples

A rational exponent is a fractional power: $a^{p/q}$ equals the $q$-th root of $a^p$. This article covers the conversion between rational exponents and radicals, three worked examples, the common slips, and a full cheat sheet of exponent rules — product, quotient, power-of-a-power, zero, negative, and rational — in one reference table.

Complement of a Set — Definition, Symbol, Examples

The complement of a set $A$, written $A'$ (or $A^c$, or $\overline{A}$), is the set of all elements in the universal set $U$ that are not in $A$. This article covers the definition, the symbol, three worked examples, the properties, and the common slips students make when working with complements.

Intersection of Sets — Symbol, Definition, Examples

The intersection of sets $A$ and $B$, written $A \cap B$, is the set of elements that are in both $A$ and $B$. This article covers the formal definition, the symbol, properties, three worked examples, the common slips, and a side-by-side comparison with union, set difference, and complement.

Cubic Polynomials — Definition, Properties, Examples

A cubic polynomial is a polynomial of degree 3, of the form $p(x) = ax^3 + bx^2 + cx + d$ with $a \neq 0$. This article covers the definition, the shape and roots of its graph, three worked examples, the most common slips, and the 16th-century rivalry between del Ferro, Tartaglia, and Cardano that produced the cubic formula.

Multiplication of Vectors — Dot & Cross Product Explained

Two non-zero vectors can be multiplied in two ways: the dot product (a scalar), which measures how much one vector projects onto the other; and the cross product (a vector), which produces a third vector perpendicular to both with magnitude equal to the parallelogram area they span.

Roots of Quadratic Equation — Types and Formulas

The roots of a quadratic equation $ax^2 + bx + c = 0$ are the values of $x$ that satisfy it. The discriminant $D = b^2 - 4ac$ classifies them into three types — real and distinct, real and equal, or complex. This article covers the quadratic formula, Vieta's sum-and-product relations, three worked examples, and a 1,400-year history from Brahmagupta to Cardano.

Reflexive Property of Equality — Definition and Examples

The reflexive property of equality says every quantity equals itself — $a = a$ for every $a$. This article covers the definition, three worked examples (algebra, geometry, relations), the most common slips, and a full side-by-side comparison with the other properties of equality (symmetric, transitive, addition, subtraction, multiplication, division, substitution).

Zero Product Property — Definition, Formula, Examples

The zero product property states that if a product of factors equals zero, at least one factor must be zero. This article covers the formal statement, a four-step method, three worked examples at three difficulty tiers, an examples-by-equation-type reference table, a real-world word problem, the advantages and limits of the property, the common slips, and the cases where the property does not apply.

Set Builder Notation — Rules and Worked Examples

Set builder notation describes a set by stating the rule its elements obey — written as ${x \mid \text{condition on } x}$. This article covers the syntax, three worked examples, a side-by-side comparison with roster notation, and the small list of symbols you need to read it fluently.

Basics of Algebra — Variables, Expressions, Equations

The basics of algebra are five objects — variables, constants, expressions, equations, and inequalities — combined by the four arithmetic operations under one balance rule. This article walks through each foundational concept with three worked examples, the most common beginner errors, and the 9th-century origin of the word algebra itself.

Substitution Method for Systems of Equations

The substitution method solves a system of two linear equations by isolating one variable in one equation, plugging that expression into the other equation, then back-substituting. This article walks through the five steps, three worked examples at increasing difficulty, where students lose marks, and how substitution stacks up against elimination and graphing.

Cube Root of 27 — Simplified Form, Methods, and Proof

The cube root of 27 is exactly $\sqrt[3]{27} = 3$, because $3 \times 3 \times 3 = 27$. $27$ is a perfect cube, so the cube root is a clean integer — no decimal approximation needed. This article covers the value, the prime-factorisation method, where $\sqrt[3]{27}$ shows up, and the slips students make most often.

Cube Root of 343 — Simplified Form and How to Find It

The cube root of 343 is exactly $\sqrt[3]{343} = 7$, because $7 \times 7 \times 7 = 343$. $343$ is a perfect cube, so the cube root is a clean integer — no decimal approximation needed. This article covers the value, the prime-factorisation method, where $\sqrt[3]{343}$ shows up, and the slips students make most often.

Square Root of 61 — How to Find It

The square root of 61 is approximately $\sqrt{61} \approx 7.8102$ — irrational, non-terminating, and cannot be simplified into a cleaner radical form because $61$ is prime. This article covers the value in exact and decimal form, three methods to compute it, where $\sqrt{61}$ shows up, and the slips students make most often.

Square Root of 85 — Value, Method & How to Find

The square root of 85 is approximately $\sqrt{85} \approx 9.2195$ — irrational, non-terminating, and cannot be simplified into a cleaner radical form. This article covers the value in exact and decimal form, three methods to compute it, where $\sqrt{85}$ shows up, and the slips students make most often.

Vertical Line Test — How to Tell if a Graph Is a Function

The vertical line test says a graph represents a function if and only if no vertical line crosses it at more than one point. Draw a vertical line anywhere on the graph — if it hits the curve once (or zero times), keep going. If it hits twice, the relation is not a function. This article covers the rule, the pass-vs-fail table for common graphs, three worked examples, and the mistakes that quietly cost marks.

Exponential Form — Converting from Logarithmic & Radical

Exponential form writes repeated multiplication as a base raised to an exponent — for example $8 = 2^3$ instead of $2 \times 2 \times 2$. This article covers what exponential form is, how to convert to and from logarithmic and radical forms, the conversion-pattern table that captures every move on one page, three worked examples, and the mistakes that quietly cost marks.

Negative Exponents — Rules, Examples & How to Simplify

A negative exponent means take the reciprocal of the base, then apply the positive version of the exponent: $a^{-n} = \dfrac{1}{a^n}$. So $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$. This article covers the rule, the sign-pattern table that tells you what the answer looks like before you compute, three worked examples at increasing difficulty, and the mistakes that quietly cost marks.

Types of Functions — Complete Classification with Examples

The types of functions are organised into three groups: by mapping (how inputs connect to outputs — one-one, onto, into, bijection), by degree (constant, linear, quadratic, cubic, polynomial), and by concept. This article walks through every type students meet from middle school to early college, with one-line examples and graphs where relevant.

Sigma Notation (Σ) - Summation Formula & Examples

Sigma notation uses the Greek capital letter $\Sigma$ to write a long sum compactly — for example, $\sum_{i=1}^{10} i$ means $1 + 2 + 3 + \dots + 10 = 55$. This article covers how to read sigma notation, the index conventions, the four core summation formulas, three worked examples at increasing difficulty, and the mistakes that quietly cost marks.

X Squared (x²) - Meaning, Graph, and Properties

x squared — written $x^2$ — means $x$ multiplied by itself. If $x = 3$, then $x^2 = 9$. This article covers what $x^2$ means, what its graph looks like (a parabola), the difference between $x^2$ and $2x$, the key properties of squaring, and a quick-reference power table for the small values you will use most.

Factored Form - Definition, Conversion, and Examples

The factored form of a quadratic writes it as a product of linear factors — for example $x^2 + 5x + 6$ becomes $(x+2)(x+3)$. This article covers what factored form is, how it compares to standard form and vertex form, how to convert between them, three worked examples at increasing difficulty, and the mistakes that quietly cost marks.

Trinomial - Definition, Types, and Factoring Methods

A trinomial is a polynomial with exactly three unlike terms — most commonly written as $ax^2 + bx + c$. This article covers the definition, the three types of trinomials you will meet at school (quadratic, perfect square, cubic), the four factoring methods compared side by side, three worked examples at increasing difficulty, and the mistakes that quietly cost marks.

Exponential Equations — Solving Methods, Formulas & Examples

An exponential equation is an equation where the variable sits in the exponent — like $2^x = 32$ or $3^{2x+1} = 81$. This article covers what makes an equation exponential, the three methods used to solve them (same base, logarithms, substitution), three worked examples at increasing difficulty, and the mistakes that quietly cost marks.

Fractional Exponents — Rules, Examples & How to Simplify

A fractional exponent is a root written as a power — $a^{1/n}$ means the $n$-th root of $a$, and $a^{m/n}$ means take the $n$-th root and raise it to the $m$-th power. This article covers the rules that govern fractional exponents, three worked examples at increasing difficulty, the three mistakes that quietly cost marks, and where rational powers turn up beyond the algebra classroom.

Cube Root of 64 - Value, Method, Examples

The cube root of 64 is exactly $4$, since $4^3 = 4 \times 4 \times 4 = 64$. Because $64$ is a perfect cube, its cube root is a whole number — no decimal, no approximation. The simplified radical form is $\sqrt[3]{64} = 4$.

Expression in Math - Definition, Types, Examples

A mathematical expression is a combination of numbers, variables, and operations that represents a value — but contains no equals sign. Examples: $3x + 5$, $2y^2 - 7$, $\sqrt{x + 1}$. An expression with an equals sign attached becomes an equation.

Determinant of a Matrix - Formula and Calculation

The determinant of a square matrix is a single number that captures the matrix's area-scaling factor (or volume-scaling in 3D), its invertibility (non-zero $\Leftrightarrow$ invertible), and the orientation it produces (positive or negative). For $2\times 2$: $\det = ad - bc$. For $3 \times 3$: cofactor expansion.

Adjoint of a Matrix - Formula, Steps, Examples

The adjoint of a matrix $A$ (also called adjugate) is the transpose of the cofactor matrix. It's the bridge that connects the matrix to its inverse: $A^{-1} = \dfrac{1}{\det(A)} \operatorname{adj}(A)$.

Inverse of 2x2 Matrix - Formula, Steps, Examples

The inverse of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$ is given by a clean formula: $A^{-1} = \dfrac{1}{ad - bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix}$. The denominator $ad - bc$ is the determinant — if it's zero, the inverse doesn't exist.

Solving an Equation - Linear, Quadratic, Radical

Solving an equation means finding the value (or values) of the unknown that make both sides equal. The four-step universal process — simplify, isolate the variable, undo operations in reverse order, check — applies to every equation type.

How to Solve for x in Algebraic Equations

To solve for $x$ means to isolate $x$ on one side of an equation using legal moves. The method depends on the equation type — linear, quadratic, radical, rational, or exponential — but the underlying logic is always the same: undo operations in reverse order until $x$ stands alone.

Discriminant in Quadratic Equations - Formula & Examples

The discriminant $\Delta = b^2 - 4ac$ is the piece under the square root in the quadratic formula. Its sign tells you the nature of a quadratic's roots before you finish solving — two distinct real roots ($\Delta > 0$), one repeated real root ($\Delta = 0$), or two complex roots ($\Delta < 0$).

Properties of Logarithms - Laws, Formulas, Proofs

The properties of logarithms are seven identities that simplify logarithmic expressions: 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 ($\log_b(x^n) = n \log_b x$), change-of-base formula, and three special values ($\log_b 1 = 0$, $\log_b b = 1$, $\log_b b^n = n$).

Axis of Symmetry - Formula, Equation, Examples

The axis of symmetry of a parabola is the vertical line that splits the curve into two mirror-image halves. For a parabola in standard form $y = ax^2 + bx + c$, the equation of the axis of symmetry is $x = -\frac{b}{2a}$. In vertex form $y = a(x - h)^2 + k$, the axis is simply $x = h$.

Arithmetic Sequence - Formula, Definition, Examples

An arithmetic sequence (or arithmetic progression, AP) is an ordered list of numbers where the difference between consecutive terms is constant — that constant is called the common difference ($d$). The nth-term formula is $a_n = a_1 + (n-1)d$ and the sum of the first $n$ terms is $S_n = \frac{n}{2}(a_1 + a_n)$.

Exponent Rules - Laws of Exponents, Formulas, Examples

The eight core exponent rules simplify expressions involving powers: the product rule, the quotient rule, the power rule, and supporting rules for the power of a product, power of a quotient, zero exponent, negative exponent, and fractional exponent. This article gives the laws with worked examples, the historical origin, the entities involved, and the most common student mistakes.

Algebraic Expression - Parts, Types, and Examples

An algebraic expression is a mathematical phrase combining variables, constants, and operations — but with no equals sign. This article covers the formal definition, every part, the four types, simplification rules, the most common student mistakes, and where the entities live in modern math.

Square Root of 12 - Value, Simplification, Examples

The square root of 12 ($\sqrt{12}$) equals $2\sqrt{3}$ in simplest radical form, or approximately $3.464$ in decimal. This article covers the simplification step ($12 = 4 \times 3$, so $\sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}$), the long-division decimal computation, where $\sqrt{12}$ appears in geometry, and the common mistakes when simplifying radicals.

Square Root of 64 - Value, Calculation, Examples

The square root of 64 ($\sqrt{64}$) equals exactly $8$ — no decimal expansion, no irrationality, just a clean whole number. This article covers why $\sqrt{64} = 8$, three methods to compute it, where 64 shows up as a perfect square in computing and geometry, and the most common mistakes.

Square Root of 10 - Value, Calculation, Examples

The **square root of 10** ($\sqrt{10}$) is approximately $3.162$. This article gives the exact form, the decimal approximation to several places, the long-division method to compute it by hand, where $\sqrt{10}$ shows up in geometry (the diagonal of a $1 \times 3$ rectangle), and why it's irrational.

Linear Equations - Definition, Forms, and Graphs

A linear equation is an equation where the variables appear only to the first power - no squares, no cubes, no roots. The graph is always a straight line. This article covers the three standard forms, how to graph each, how to convert between them, and the most common solving mistakes.

Logarithm Rules - Product, Quotient, and Power

Three logarithm rules turn complicated arithmetic into simple arithmetic: the product rule ($\log_b(xy) = \log_b x + \log_b y$) turns multiplication into addition; the quotient rule turns division into subtraction; the power rule turns exponents into multiplication.

Multiplication of Matrices - Rules and Methods

The multiplication of matrices is defined only when the number of columns in the first matrix equals the number of rows in the second — and the result is computed row-by-column using dot products. This article covers the dimension rule, the step-by-step method, the four properties, worked examples in $2 \times 2$ and $3 \times 3$ cases and the most common mistakes.

Coefficient - Definition, Types, and Math Examples

A coefficient is the number that multiplies a variable in an algebraic expression — in $3x$, the coefficient is $3$. This article covers the formal definition, the four types of coefficients, how to identify them in any expression, the rule for variables with no visible number, and the most common mistakes when picking out the coefficient.

Sequences in Algebra - Types, Formulas, Examples

A sequence in algebra is an ordered list of numbers — called terms — that follows a specific rule. Sequences can be finite (ending) or infinite (continuing forever). The eight most-studied types are arithmetic (add a constant), geometric (multiply by a constant), harmonic (reciprocals of arithmetic), Fibonacci (each term = sum of previous two), triangular, square, cube, and quadratic.

Sets in Math - Definition, Types, and Symbols

A set in math is a well-defined collection of distinct objects, called elements, enclosed in curly braces — for example, $A = {2, 4, 6, 8}$. This article covers the formal definition, the 12 most common types of sets (finite, infinite, empty, singleton, equal, equivalent, subset, superset, universal, power, disjoint, overlapping), the symbols used to describe them ($\in$, $\subset$, $\cup$, $\cap$, $\varnothing$), the four core operations, and the mistakes that trip up most students.

Quadratic Equations - Formula, Solving, Examples

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$ where $a \neq 0$. This article covers the standard form, the three methods for solving (factoring, completing the square, and the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$).

Equation in Math - Definition, Types, and Parts

An equation in math is a statement that two expressions are equal, separated by an equals sign — for example, $3x + 5 = 14$. This article covers the formal definition, every part of an equation, the seven main types you'll meet in school, and the most common mistakes when solving them.

Algebra - Concepts, Topics, and Basic Formulas

Algebra is the branch of mathematics that uses letters and symbols — usually $x$, $y$, $z$ — to represent unknown numbers, and then works out what those numbers must be. This article covers the core concepts, the main topics, the basic formulas you'll use again and again.

Perfect Squares - List of Squares and Formula

Polynomials – Definition, Types, and Formulas