Book A Free Math Class

Topic

Algebra

256 articles

math Objective Function — Definition, Formula, and Examples \ \ The objective function is the linear expression a linear programming problem sets out to maximize or minimize — usually written $Z = ax + by$, where $x$ and $y$ are decision variables. This article defines it, shows the $Z = ax + by$ form, works six examples, and separates it clearly from the constraints.\ \ Algebra math Scalar Triple Product — Definition, Formula, Example \ \ The scalar triple product of three vectors is $\vec{a} \cdot (\vec{b} \times \vec{c})$, a single number equal to the signed volume of the parallelepiped built on those vectors. This article covers the definition, the determinant formula, six worked examples, why a zero result means the vectors are coplanar, and the mistakes to avoid\ \ Algebra math Square Root 1 to 25 — Values, Chart & Examples \ \ The square root 1 to 25 chart gives $\sqrt{1}$ through $\sqrt{25}$, where five values ($\sqrt{1}, \sqrt{4}, \sqrt{9}, \sqrt{16}, \sqrt{25}$) are exact whole numbers and the other twenty are irrational, rounded here to three decimals. This article gives the full table, marks rational versus irrational, and shows how to read and rebuild it.\ \ Algebra math Square Root of 43 — Value, Simplification, and Steps \ \ The square root of 43 is approximately $6.557$ and cannot be simplified, because $43$ is a prime number, so $\sqrt{43}$ is already in simplest form. This article gives the value, shows why it is irrational, and works the long-division method step by step.\ \ Algebra math Square Root of 105 — Value and How to Find It \ \ The square root of 105 is $\sqrt{105} \approx 10.247$, an irrational number that cannot be simplified because $105 = 3 \times 5 \times 7$ has no repeated prime factor. This article shows the factorization check, the long-division method, where it appears, common mistakes, and worked examples.\ \ Algebra math Square Root of 47 — Value, Simplification, and Steps \ \ The square root of 47 is approximately 6.856, and it is irrational because 47 is a prime number with no square factors. This article shows why $\sqrt{47}$ is already in its simplest radical form, how to compute it by long division and estimation, and the errors to avoid.\ \ Algebra math Square Root of 115 — Value, Simplification, and Steps \ \ The square root of 115 is approximately $10.724$, and it is irrational because $115 = 5 \times 23$ has no perfect-square factor. This article gives the value, shows why $\sqrt{115}$ is already in simplest radical form, and works the long-division method step by step.\ \ Algebra math Square Root of 153 — Value, Simplification, and Steps \ \ The square root of 153 is $3\sqrt{17}$ in exact form and approximately $12.369$ as a decimal. This article gives the value, shows why $\sqrt{153}$ is irrational, simplifies it by prime factorization, and works the long-division method step by step.\ \ Algebra math Square Root of 157 — Value and How to Find It \ \ The square root of 157 is $\sqrt{157} \approx 12.530$, an irrational number that cannot be simplified because 157 is prime. This article shows why the radical stays as is, the long-division method, where it appears, common mistakes, and worked examples.\ \ Algebra math Square Root of 245 — Value, Simplification, Steps \ \ The square root of 245 simplifies to 7√5 ≈ 15.652, because 245 factors as 5 × 7² and the perfect square 49 pulls out as 7. This article gives the value, the full simplification, two computation methods, and the mistakes students make with √245.\ \ Algebra math Square Root of 255 — Value, Simplification, and Steps \ \ The square root of 255 is approximately $15.969$ and cannot be simplified, because $255$ has no perfect-square factor, so $\sqrt{255}$ is already in simplest form. This article gives the value, shows why it is irrational, and works the long-division method step by step\ \ Algebra math Square Root of 340 — Value, Simplification, Steps \ \ The square root of 340 simplifies to 2√85 ≈ 18.439, because 340 factors as 2² × 5 × 17, and only the perfect square 4 pulls out. This article gives the value, the full simplification, two hand methods, and the mistakes students hit with √340.\ \ Algebra math Square Root of 384: Value and Simplification \ \ The square root of 384 simplifies to $8\sqrt{6} \approx 19.596$, an irrational number because $384 = 2^7 \times 3$ leaves factors that cannot pair up perfectly. This article shows the simplification, the long-division estimate, and where the value comes from.\ \ Algebra math Square Root of 640 — Value, Simplification, and Steps \ \ The square root of 640 is $8\sqrt{10}$, about 25.298, and it is irrational because 640 is not a perfect square. This article shows how to pull the perfect-square factor 64 out of the radical, compute the decimal by long division, and avoid the usual simplification slips\ \ Algebra math Square Root of 567 — Value, Simplification, and Steps \ \ The square root of 567 is $9\sqrt{7}$ in exact form and approximately $23.812$ as a decimal. This article gives the value, shows why $\sqrt{567}$ is irrational, simplifies it by prime factorization, and works the long-division method step by step.\ \ Algebra math Square Root of 4900 — Value, Steps, and Why It Is 70 \ \ The square root of 4900 is exactly 70, because $70 \times 70 = 4900$. Since 4900 is a perfect square, its root is a whole, rational number, and this article shows three ways to reach 70 plus the mistakes that make students miss it.\ \ Algebra math Squaring a Trinomial: Formula and Examples \ \ Squaring a trinomial applies the identity $(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca$, three squares plus twice each pair-product. This article derives the formula, works through examples with numbers and variables, and shows why the cross terms are the part everyone forgets.\ \ Algebra math Sum of Odd Numbers — Formula, Proof & Examples \ \ The sum of odd numbers follows one clean rule: the first $n$ odd numbers add up to $n^2$. This article gives the formula $1 + 3 + 5 + \dots + (2n-1) = n^2$, a visual square-building proof, six worked examples, and the mistakes that trip students up.\ \ Algebra math Synthetic Division of a Polynomial — Steps and Examples \ \ Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form $x - c$, using only the coefficients instead of the full variables. This article defines the method, walks six worked examples including a wrong-path check, explains why it works through the Remainder Theorem, and names the mistakes that trip students up.\ \ Algebra math Zeros of a Polynomial — Definition & Examples \ \ The zeros of a polynomial are the values of $x$ that make the polynomial equal to zero — the solutions of $p(x) = 0$, also called its roots. This article defines them, shows how to find them, works six examples, and draws the sharp line between zeros of a polynomial and the zero polynomial.\ \ Algebra math Applications of Linear Equations — Real-World Uses \ \ An application of a linear equation is any real-world problem you solve by translating words into an equation of the form $ax + b = c$ and solving for the unknown. This article covers the translation method, six worked applications, the mistakes that trip students up, and where these equations show up in daily life.\ \ Algebra math Augmented Matrix: Definition and Examples \ \ An augmented matrix writes a system of linear equations as a single grid $[A \mid b]$, the coefficient matrix $A$, a vertical bar standing in for the equals sign, and the constants $b$. This article shows how to build one from a system, how row operations solve it, and the mistakes that scramble the setup.\ \ Algebra math Cube Root of 3 — Value, Steps, and Estimation \ \ The cube root of 3 is $\sqrt[3]{3} \approx 1.442$, an irrational number that cannot be simplified because 3 is prime. This article shows why the radical stays as is, how to estimate its value by hand, where it appears, common mistakes, and worked examples.\ \ Algebra math Cube Root of 25: Value and Steps \ \ The cube root of 25 is $\sqrt[3]{25} \approx 2.924$, an irrational number because $25 = 5^2$ has no factor repeated three times. This article gives the value, shows why it will not simplify, and walks through two ways to estimate it by hand.\ \ Algebra math Cube Root of 5 — Value and Steps \ \ The cube root of 5 is $\sqrt[3]{5} \approx 1.710$, an irrational number that cannot be simplified because 5 is prime. This article shows why the radical stays as is, how to estimate its value by hand, where it appears, common mistakes, and worked examples.\ \ Algebra math Cube Root of 100 — Value and Steps \ \ The cube root of 100 is an irrational number equal to ∛100 ≈ 4.642, because 100 factors as 2² × 5² with no perfect cube inside it. This article gives the value, the reason ∛100 cannot be simplified, two ways to compute it by hand, and the mistakes to avoid.\ \ Algebra math Determinants — Definition, Formula, and Examples \ \ A determinant is a single number computed from a square matrix that tells you whether the matrix can be inverted and how it scales area or volume. This article defines determinants, works the 2×2 and 3×3 expansions step by step, lists the properties that speed the arithmetic up, and shows where the answer matters.\ \ Algebra math Floor and Ceiling Function — Definition and Examples \ \ The floor function $\lfloor x \rfloor$ gives the greatest integer less than or equal to $x$, and the ceiling function $\lceil x \rceil$ gives the least integer greater than or equal to $x$. This article covers their notation, graphs, properties, and worked examples, plus the negative-number trap that costs the most marks.\ \ Algebra math Linear, Quadratic, and Cubic Polynomials \ \ Linear, quadratic, and cubic polynomials are the first three families in the degree ladder: degree 1 ($ax+b$), degree 2 ($ax^2+bx+c$), and degree 3 ($ax^3+bx^2+cx+d$). This article covers how to classify by degree, each general form, the graph each produces, and the mistakes that trip students up.\ \ Algebra math Multiplying Polynomials — Methods, Steps, and Examples \ \ Multiplying polynomials means multiplying every term of one polynomial by every term of the other, then combining like terms. This article covers the distributive, FOIL, box, and vertical methods, works through six examples, and fixes the exponent and sign mistakes that lose marks.\ \ Algebra math Square Root of 441 — Value, Simplification, and Steps \ \ The square root of 441 is exactly 21, because $21 \times 21 = 441$. This article shows why 441 is a perfect square, works the prime-factorisation and long-division methods, gives a quick-reference table, and points to where $\sqrt{441}$ appears.\ \ Algebra math Square Root of 1369 — Value, Simplification, and Steps \ \ The square root of 1369 is exactly 37, because $37 \times 37 = 1369$. This article shows why 1369 is a perfect square, works the prime-factorisation and long-division methods, gives a quick-reference table, and points to where $\sqrt{1369}$ appears.\ \ Algebra math Square Root of 1000 — Value, Simplification, and Steps \ \ The square root of 1000 is $10\sqrt{10}$ in simplified radical form, which equals approximately $31.623$. This article shows how prime factorization strips 1000 down to $10\sqrt{10}$, how to estimate the decimal by hand, and where this value turns up.\ \ Algebra math Square Root of 2025 — Value, Simplification, and Steps \ \ The square root of 2025 is exactly 45, because $45 \times 45 = 2025$. This article shows why 2025 is a perfect square, works the prime-factorisation and long-division methods, gives a quick-reference table, and points to where $\sqrt{2025}$ shows up\ \ Algebra math Square Root of 2000: Value and Simplified Radical Form \ \ The square root of 2000 is $20\sqrt{5}$ in simplest radical form, which is approximately $44.721$. Because $2000$ is not a perfect square, $\sqrt{2000}$ is irrational; this article shows the simplification steps, the estimation method, and common mistakes.\ \ Algebra math Square Root of 3600 — Value & Steps \ \ The square root of 3600 is exactly 60, because $60 \times 60 = 3600$. This article shows why 3600 is a perfect square, works the prime-factorisation and long-division methods, gives a quick-reference table, and points to where $\sqrt{3600}$ appears.\ \ Algebra math Standard Form — Definition, Types, and Examples \ \ Standard form is the agreed, tidy way of writing a mathematical object — a number as $a \times 10^n$ (scientific notation), a polynomial with degrees in descending order, or an equation in a fixed layout. This article covers all three meanings, six worked examples, and why the phrase means different things in the US and UK.\ \ Algebra math Standard Form of Linear Equations — Formula and Examples \ \ The standard form of a linear equation in two variables is $Ax + By = C$, where $A$, $B$, and $C$ are integers and $A$ is not negative. This article gives the definition, the rules for the coefficients, conversions to and from slope-intercept form, common mistakes, and six worked examples.\ \ Algebra math Superset — Definition, Symbol, and Examples \ \ A superset is a set that contains every element of another set: if every element of $A$ is in $B$, then $B$ is a superset of $A$, written $B \supseteq A$. This article covers the superset symbol, proper versus improper supersets, and how superset mirrors subset — with worked examples.\ \ Algebra math Zero Polynomial — Definition, Degree, and Examples \ \ The zero polynomial is the polynomial whose every coefficient is $0$, so it equals $0$ for every input and is written $P(x) = 0$. Its degree is undefined (not $0$), it has infinitely many zeros, and this article shows why through worked examples and the mistakes students make.\ \ Algebra math Cube Numbers — Definition, List, and Examples \ \ A cube number is what you get when you multiply an integer by itself three times, such as $4^3 = 64$. This article gives the definition, the list from $1^3$ to $10^3$, the patterns that generate them, negative cubes, common mistakes, and six worked examples.\ \ Algebra math Antilog Table: How to Read and Use It (With Examples) \ \ An antilog table reverses a logarithm: given $\log x = y$, it recovers $x = \text{antilog}(y)$. This article shows how to split a logarithm into its characteristic and mantissa, read the row, column, and mean-difference values, place the decimal point, and handle negative (bar) characteristics.\ \ Algebra math Cube Root of 1 — Value and Cube Roots of Unity \ \ The cube root of 1 is $\sqrt[3]{1} = 1$, because $1^3 = 1$. Yet the equation $x^3 = 1$ has three cube roots of unity — $1$, $\omega$, and $\omega^2$ — and this article gives all three values, the derivation, where they appear, and worked examples\ \ Algebra math Cube Root of 24 — Value and Simplification Steps \ \ The cube root of 24 simplifies to $\sqrt[3]{24} = 2\sqrt[3]{3}$, with a decimal value of about $2.884$. This article shows the prime-factorization simplification, the estimation method, where it appears, common mistakes, and worked examples.\ \ Algebra math Difference of Sets: Definition, Formula, and Examples \ \ The difference of sets $A - B$ is the set of elements that belong to $A$ but not to $B$: $A - B = {x : x \in A \text{ and } x \notin B}$. This article shows the formula, the Venn diagram, worked examples, and why $A - B \neq B - A$.\ \ Algebra math Division Property of Equality — Formula & Examples \ \ The division property of equality states that dividing both sides of a true equation by the same non-zero number keeps it true, so $5x = 25$ becomes $x = 5$. This article gives the formula, six worked examples, why the "non-zero" rule exists, and the errors students make most.\ \ Algebra math Explicit Function — Definition, Formula & Examples \ \ An explicit function is one where the dependent variable is solved for and written by itself, in the form $y = f(x)$, so you can compute the output straight from the input. This article gives the definition, contrasts explicit with implicit form, works six examples, and names the mistakes students make when converting between the two.\ \ Algebra math Factoring Methods — Definition, Types, and Examples \ \ Factoring methods are the standard techniques for rewriting a polynomial as a product of simpler factors: pulling out the greatest common factor (GCF), factoring by grouping, factoring trinomials, and the difference of squares pattern $a^2 - b^2 = (a+b)(a-b)$.\ \ Algebra math Graphing Linear Equations: Methods, Steps, and Examples \ \ Graphing linear equations means plotting the straight line that satisfies an equation like $y = mx + b$. This article walks through the three main methods — slope-intercept, table of values, and intercepts — with worked examples and the mistakes that produce a crooked or misplaced line.\ \ Algebra math Linear Programming: Objective, Constraints & Graph Method \ \ Linear programming finds the best value of a linear objective function subject to linear constraints, and the optimum always sits at a corner (vertex) of the feasible region. This article defines each term, walks through formulating a problem, and solves a two-variable problem with the graphical corner-point method, plus common mistakes.\ \ Algebra math Inverse of Diagonal Matrix — Formula & Examples \ \ The inverse of a diagonal matrix is found by replacing each diagonal entry with its reciprocal, so $\operatorname{diag}(d_1, d_2, \ldots, d_n)$ inverts to $\operatorname{diag}(1/d_1, 1/d_2, \ldots, 1/d_n)$. This article gives the formula, a short proof, six worked examples, and the one condition that makes the inverse fail.\ \ Algebra math Multiplication of Algebraic Expressions: Rules & Examples \ \ Multiplying algebraic expressions means using the distributive property to multiply every term of one expression by every term of the other, then combining like terms. This article covers the sign and exponent rules, monomial and binomial and polynomial products, the FOIL shortcut, worked examples, and the mistakes that cost the most marks.\ \ Algebra math Modulus of a Complex Number — Formula and Examples \ \ The modulus of a complex number $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$, the distance from the origin to the point $(a, b)$ on the complex plane. This article defines the modulus, works through examples, covers its key properties, and shows where students go wrong.\ \ Algebra math Non-Singular Matrix — Definition, Properties, and Examples \ \ A non-singular matrix is a square matrix whose determinant is not zero ($\det A \neq 0$), which means it is invertible. This article defines the non-singular matrix, contrasts it with a singular matrix, lists its properties, shows how to test a matrix, and works through 2×2 and 3×3 examples.\ \ Algebra math Prove That Root 7 Is Irrational — Proof by Contradiction \ \ To prove that root 7 is irrational, assume the opposite — that $\sqrt{7} = \frac{p}{q}$ in lowest terms — and derive a contradiction from the fact that 7 must then divide both $p$ and $q$. This article walks the full proof by contradiction, explains the prime-divisibility step it depends on, and shows the common errors\ \ Algebra math Square Root of 68 — Value & Simplified Form (2√17) \ \ The square root of 68 is $2\sqrt{17}$ in simplified radical form and approximately 8.246 as a decimal. This article gives the exact value, the prime-factorisation and long-division methods, a quick-reference table, and where $\sqrt{68}$ appears.\ \ Algebra math Simultaneous Equations — Methods and Examples \ \ Simultaneous equations are two or more equations that share the same variables and are solved together to find values that satisfy all of them at once. This article covers the substitution, elimination, and graphing methods, when each is best, common mistakes, and six worked examples.\ \ Algebra math Square Root of 78 — Value, Simplification, and Steps \ \ The square root of 78 is irrational and approximately 8.832, and √78 is already in simplest radical form because 78 has no perfect-square factor. This article shows the estimation and long-division methods, gives a quick-reference table, and points to where $\sqrt{78}$ appears.\ \ Algebra math Square Root of 360 — Value, Simplification, and Steps \ \ The square root of 360 is $6\sqrt{10}$ in simplified radical form, which equals approximately $18.974$. This article shows how to pull the perfect-square factor out of 360, how to estimate the decimal by hand, and where the value comes up.\ \ Algebra math Square Root of 260 — Value, Simplification, and Steps \ \ The square root of 260 is $2\sqrt{65}$ in simplest radical form and approximately 16.125 as a decimal. This article shows how to simplify $\sqrt{260}$ by prime factorization, estimate it between whole numbers, and explains why $\sqrt{260}$ is an irrational number.\ \ Algebra math Partial Fractions - Decomposition Method, Examples \ \ Partial fractions break a single complicated fraction like $\frac{4x + 12}{x^2 + 4x}$ into a sum of simpler ones, $\frac{3}{x} + \frac{1}{x + 4}$, the exact reverse of adding fractions over a common denominator. This article covers the cases (distinct linear, repeated linear, irreducible quadratic), the decomposition steps, and the mistakes to avoid.\ \ Algebra math Onto Function — Definition, Formula, Graph, and Examples \ \ An onto function (also called a surjective function) is one where every element of the codomain is the output of at least one input — nothing in the target set is left unreached. This article gives the formal definition, the test (range equals codomain), the formula for the number of onto functions, the onto-vs-into contrast, and six worked examples\ \ Algebra math Zero Function — Definition, Graph, Examples \ \ The zero function is the constant function $f(x) = 0$ that sends every input to the single output zero. This article gives its definition, its graph (the x-axis itself), its properties (constant, both even and odd, derivative zero), the crucial difference between the zero function and the zeros of a function, and six worked examples\ \ Algebra math Universal Set — Definition, Symbol & Examples \ \ A universal set, written $U$, is the set that contains every element under consideration for a given problem; every other set in that problem is a subset of it. This article defines $U$, fixes its symbol and Venn-diagram rectangle, separates it cleanly from a union of sets, and works through examples including how the complement of a set depends on the $U$ you choose.\ \ Algebra math Unit Matrix - Definition, Identity Matrix Link, Examples \ \ A unit matrix is a square matrix with 1s along its main diagonal and 0s everywhere else — it is simply another name for the identity matrix, written $I$ or $I_n$. This article focuses on why the two terms are the same object, how "unit" connects to the multiplicative identity, where each name is preferred, and worked examples; the full property list lives in the identity matrix article.\ \ Algebra math Unitary Matrix - Definition, Properties, Examples \ \ A unitary matrix is a square matrix of complex numbers whose conjugate transpose equals its inverse — that is, $U^H U = U U^H = I$. This article covers the definition, the conjugate-transpose test, every property (orthonormal columns, $|\det U| = 1$, eigenvalues on the unit circle), how unitary generalises the orthogonal matrix to complex numbers, and worked examples\ \ Algebra math Types of Sets — Definition, Examples & Symbols \ \ A set is a well-defined collection of distinct objects, and its type is decided by how many elements it holds and how those elements relate to other sets. The main types of sets are finite, infinite, empty, singleton, equal, equivalent, subset, universal, and power set. This article defines each one with its symbol, draws the Venn-diagram picture, works through examples, and flags the comparisons students mix up most.\ \ Algebra math Transpose of a Matrix - Definition, Properties, Example \ \ The transpose of a matrix is the matrix you get by turning its rows into columns, written $A^T$ — the entry in row $i$, column $j$ moves to row $j$, column $i$. This article covers the definition, how the order flips, all the properties (including the reversal rule $(AB)^T = B^T A^T$), the link to symmetric matrices, and six worked examples\ \ Algebra math Scalar Product — Definition, Formula, Examples \ \ The scalar product of two vectors — also called the dot product — multiplies them to give a single number, not a vector: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$. This article covers both the geometric and component formulas, the key properties, how it differs from the cross product, six worked examples, and the mistakes students make most.\ \ Algebra math Quadratic Expressions - Definition, Forms, Examples \ \ A quadratic expression is any expression of the form $ax^2 + bx + c$ where $a \neq 0$, a variable raised to the power two, plus optional lower terms. This article covers its standard form, the parabola it graphs, how to factor it, and the key distinction between a quadratic expression and a quadratic equation\ \ Algebra math Power Set — Definition, Cardinality, and Examples \ \ The power set of a set $A$, written $P(A)$, is the set of all subsets of $A$, including the empty set $\varnothing$ and $A$ itself. This article defines $P(A)$, proves the cardinality rule $|P(A)| = 2^n$ for a set of $n$ elements, lists the properties, and works through examples from the empty set up to four-element sets.\ \ Algebra math Properties of Matrices - Addition, Multiplication, Transpose \ \ The properties of matrices are the rules that govern the four core operations — addition, scalar multiplication, multiplication, and transpose — including which laws carry over from ordinary numbers and which break. This article covers each property with its formula, the big exception (matrix multiplication is not commutative: $AB \neq BA$), and six worked examples that test the rules\ \ Algebra math Polynomial Equations - Definition, Types, Examples \ \ A polynomial equation sets a polynomial equal to zero — like $x^3 - 6x^2 + 11x - 6 = 0$, and its solutions are the values of the variable that make it true. This article covers the types by degree, the link between degree and number of roots, and the methods for finding those roots.\ \ Algebra math Pascal's Triangle - Definition, Patterns, Examples \ \ Pascal's Triangle is a triangular array where every number is the sum of the two directly above it, and each row holds the coefficients of a binomial expansion. This article shows how to build the triangle, the patterns hiding inside it, the link to $\binom{n}{r}$, and the mistakes students make first.\ \ Algebra math Arithmetic Progressions — nth Term, Sum, Examples \ \ An arithmetic progression (AP) is a sequence where each term increases by a fixed amount called the common difference $d$, like $2, 5, 8, 11, \dots$. This article covers the definition, the nth-term formula $a_n = a + (n-1)d$, both sum formulas, a quick derivation, six worked examples, and the mistakes students make most\ \ Algebra math Bijective Function — One-One and Onto, Examples \ \ A bijective function is one that is both one-one (injective) and onto (surjective) — every output is reached by exactly one input, with no collisions and nothing left out. This article gives the definition, the two conditions, the one-to-one correspondence idea, the $n!$ counting formula, the link to inverses, and six worked examples\ \ Algebra math Compound Inequality — AND, OR, Solving & Examples \ \ A compound inequality joins two inequalities with the word "and" or "or", "and" gives the intersection (values satisfying both), "or" gives the union (values satisfying either). This article shows how to solve and graph each type on a number line, writes the answers in interval notation, and flags where the AND/OR logic most often goes wrong\ \ Algebra math Factorization of Algebraic Expressions - Methods and Examples \ \ Factorization of algebraic expressions means rewriting an expression as a product of its factors — turning a sum like $x^2 + 5x + 6$ into $(x+2)(x+3)$. This article walks through the four core methods (common factor, grouping, algebraic identities, and splitting the middle term), worked examples, and the errors that trip students up\ \ Algebra math Injective Function — How to Prove It, and Examples \ \ An injective function (also called a one-one function) is one where distinct inputs always produce distinct outputs — no two domain elements share the same image. This article gives the formal definition, the algebraic proof method, the horizontal line test, the injective-vs-surjective contrast, the counting formula, and six worked examples\ \ Algebra math Inverse Functions — Definition, Steps, Examples \ \ An inverse function, written $f^{-1}$, undoes the original function — feed an output back in and you recover the original input. This article gives the definition, the four-step method to find an inverse, the verification by composition, the reflection-over-$y=x$ graph, why only bijective functions are invertible, and six worked examples.\ \ Algebra math Linear Inequalities — Definition, Rules, and Examples \ \ A linear inequality compares two linear expressions using $<$, $>$, $\leq$, or $\geq$ instead of an equals sign, so its solution is a range of values, not a single number. This article gives the solving rules, including the one rule that flips the sign, dividing or multiplying by a negative, shows how to graph the solution on a number line, and works through one- and two-variable examples.\ \ Algebra math Matrix Scalar Multiplication - Rules, Properties, Examples \ \ Matrix scalar multiplication means multiplying every entry of a matrix by a single number called a scalar — if $A = [a_{ij}]$ and $k$ is a scalar, then $kA = [k,a_{ij}]$. This article covers the rule, all its properties (commutative, associative, distributive), how it differs from multiplying two matrices, and six worked examples.\ \ Algebra math Nth Term of a GP — Formula and Worked Examples \ \ The nth term of a GP (geometric progression) is found with $a_n = ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio. This article gives you the formula, a clean derivation, the term-from-the-end version, six worked examples, and the slips that cost marks — so you can find any term without writing out the whole sequence\ \ Algebra math Subtraction Property of Equality - Definition & Examples \ \ The subtraction property of equality states that if $a = b$, then $a - c = b - c$ - subtracting the same number from both sides of an equation keeps it balanced. It is the rule that lets you isolate a variable by stripping away an added term. This article gives the formula, works six examples including fractions and a two-column proof step, and shows how this property and the addition property are two halves of the same balance principle.\ \ Algebra math Log Table: How to Read & Use a Logarithm Table \ \ A log table is a reference chart that gives the base-10 logarithm of a number, used to multiply, divide, and find powers by hand. To read a value, you split the logarithm into two parts: the characteristic (the integer part) and the mantissa (the decimal part, read from the table).\ \ Algebra math Algebra 2: Topics, Concepts & How They Connect \ \ Algebra 2 is the high-school course that extends Algebra 1 into the full world of functions (polynomial, rational, radical, exponential, logarithmic, and trigonometric) and adds matrices, sequences, and conic sections. The thread tying Algebra 2 together is the idea of a function: a rule that takes an input and returns a single output.\ \ Algebra math Constants in Math: Definition, Examples & Types \ \ A constant is a fixed value that does not change, such as the number 5, the fraction $\tfrac{1}{2}$, or a named value like $\pi$. In an expression, the constant term is the part with no variable attached.\ \ Algebra math Absolute Value: Definition, Symbol, Graph & Examples \ \ The absolute value of a number is its distance from zero on the number line, written with vertical bars as $|x|$, and it is never negative. So $|5| = 5$ and $|-5| = 5$.\ \ Algebra math Periodic Function: Definition, Formula, Graph & Examples \ \ A periodic function is a function whose values repeat at fixed, regular intervals. The length of one full repeat is the period, written $P$, and it satisfies $f(x + P) = f(x)$ for every $x$.\ \ Key terms: period (the length of one full cycle), amplitude (the height from the midline to the peak, a separate measurement), cycle (one complete repeat), midline (the horizontal line halfway between peak and trough).\ \ Algebra math Parabolic Function — Definition, Formula, Graph, and Examples \ \ A parabolic function is a degree-two function $f(x) = ax^2 + bx + c$ (with $a \neq 0$) whose graph is a U-shaped curve called a parabola. This article defines the parabolic function, draws its graph with vertex and axis of symmetry marked, derives the vertex formula, gives its domain and range, and works through six examples — including how it differs from a plain quadratic equation.\ \ Algebra math Into Function — Definition, Graph, and Examples (Into vs Onto) \ \ An into function is one where at least one element of the codomain is never reached — its range is a strict subset of its codomain. This article defines the into function, contrasts it cleanly with an onto (surjective) function, shows how to read it off arrow diagrams and graphs, and works through six examples — including why $f(x) = x^2$ over the reals is "into."\ \ Algebra math Identity Function — Definition, Graph, Properties, and Examples \ \ An identity function returns whatever you put in, unchanged — its rule is $f(x) = x$. This article defines the identity function, draws its graph as the straight line $y = x$ through the origin, lists its key properties (it is its own inverse, bijective, slope $1$), and works through six examples — including how it differs from a constant function.\ \ Algebra math Inverse Relation - Definition, Formula & Examples \ \ An inverse relation is what you get when you swap the two coordinates of every ordered pair in a relation — turning each $(x, y)$ into $(y, x)$. This article defines the inverse relation, shows how its graph reflects across the line $y = x$, proves the inverse relation theorem $(R^{-1})^{-1} = R$, and works through six examples — including why an inverse relation is not always an inverse function.\ \ Algebra math Discontinuous Function - Types, Graph & Examples \ \ A discontinuous function is one whose graph cannot be drawn without lifting your pen — it has at least one break, hole, jump, or vertical asymptote. This article walks through the three main types — removable, jump, and infinite discontinuities — shows how to read each off a graph, and works through six examples with the mistakes to avoid.\ \ Algebra math Substitution Property of Equality - Definition & Examples \ \ The substitution property of equality says that if $a = b$, then $a$ can replace $b$ (or $b$ can replace $a$) in any expression or equation without changing its value. It is the rule that makes "plugging in" a known value legitimate. This article gives the definition, works six examples across algebra, geometry, and limits, and clears up the most common confusion: how the substitution property differs from the transitive property.\ \ Algebra math Sum of an Infinite GP - Formula & Examples \ \ An infinite geometric progression has a finite sum, $S_\infty = \dfrac{a}{1 - r}$, but only when the common ratio satisfies $|r| < 1$. Here $a$ is the first term and $r$ is the common ratio; if $|r| \geq 1$ the series diverges and has no finite sum. This article explains why shrinking terms add up to a limit, derives the formula, and works examples, including how a repeating decimal like $0.999\ldots$ turns out to equal exactly $1$.\ \ Algebra math Sum of a GP - Formula, Derivation & Examples \ \ The sum of the first $n$ terms of a geometric progression is $S_n = \dfrac{a(1 - r^n)}{1 - r}$ when the common ratio $r \neq 1$, and $S_n = na$ when $r = 1$. Here $a$ is the first term and $r$ is the common ratio. This article derives the formula by the multiply-and-subtract trick, explains why the $r = 1$ case is separate, and works six examples from clean integers to a folding-paper word problem.\ \ Algebra math Cramer's Rule - Formula, 2x2 & 3x3 Examples \ \ Cramer's rule solves a system of linear equations by writing each unknown as a ratio of two determinants. For a system with coefficient determinant $D$, the value of each variable is its own determinant divided by $D$, where each numerator swaps one column of the coefficient matrix for the constants. This article gives the 2x2 and 3x3 formulas, derives where they come from, works six examples, and pins down the one condition ($D \neq 0$) that decides whether the method works at all.\ \ Algebra math Elimination Method — Steps and Worked Examples \ \ The elimination method solves a system of equations by adding or subtracting them so one variable cancels, leaving a single equation in one variable. This article covers the steps, when to add versus subtract, how to scale coefficients first, the no-solution and infinite-solution cases, and the mistakes that cost marks.\ \ Algebra math System of Equations — Types, Methods, Examples \ \ A system of equations is two or more equations sharing the same variables, and its solution is the set of values that make every equation true at once. This article covers the definition, the three solution types (one solution, no solution, infinitely many), how to solve by graphing, substitution, and elimination, and the mistakes that lose marks.\ \ Algebra math Union of Sets — Definition, Symbol, and Formula \ \ The union of two sets $A$ and $B$, written $A \cup B$, is the set of all elements that are in $A$, in $B$, or in both — with each element listed once. This article covers the symbol $\cup$, the set-builder definition, the cardinality formula $n(A \cup B) = n(A) + n(B) - n(A \cap B)$, the properties of union, and the mistakes that cost marks.\ \ Algebra math Operations on Sets — Union, Intersection, Complement \ \ Operations on sets are the rules that combine or compare sets — union $\cup$, intersection $\cap$, difference $-$, and complement $'$ — each producing a new set. This article defines all four with their symbols and Venn diagrams, lists the key properties (commutative, associative, distributive, De Morgan's laws), gives the cardinality formula, and works through common mistakes.\ \ Algebra math Empty Set — Definition, Symbol, and Properties \ \ The empty set is the one set that has no elements at all, written $\varnothing$ or ${}$, with cardinality $0$. This article covers its symbol, why there is only one empty set, why it counts as a subset of every set, how it differs from ${0}$ and from the number zero, and the mistakes that trip students up.\ \ Algebra math Eigenvalues: Definition, How to Find Them & Examples \ \ An eigenvalue is the scalar $\lambda$ by which a matrix stretches or shrinks a special vector — its eigenvector — without turning it. This article covers the defining equation $A\mathbf{v} = \lambda\mathbf{v}$, how to find eigenvalues from the characteristic equation $\det(A - \lambda I) = 0$, the trace-and-determinant properties, and the sign and factoring mistakes that derail the calculation.\ \ Algebra math Rank of a Matrix: Definition, Methods & Examples \ \ The rank of a matrix is the number of linearly independent rows (equal to the number of independent columns) it has — the true amount of information the matrix carries. This article covers the echelon-form and minor methods of finding rank, the properties (rank $\leq \min(m, n)$, rank of the transpose), and how rank tells you whether a system has a solution.\ \ Algebra math Matrix Equation: How to Solve AX = B with Examples \ \ A matrix equation writes a system of linear equations in the compact form $AX = B$, where $A$ holds the coefficients, $X$ the unknowns, and $B$ the constants. This article shows how to build the equation, solve it with $X = A^{-1}B$, decide when a unique solution exists ($\det A \neq 0$), and avoid the order and singular-matrix mistakes.\ \ Algebra math Transformation Matrix: Types, Formulas & Examples \ \ A transformation matrix is a matrix that turns one vector into another through matrix multiplication, remapping every point in space at once. This article covers what it does, the five core types — scaling, rotation, shear, reflection, and translation — their matrix forms, how to apply them, and the mistakes that flip a figure the wrong way.\ \ Algebra math Rotation Matrix: 2D & 3D Formula, Derivation, Examples \ \ A rotation matrix is a square matrix that rotates a vector around the origin while keeping its length unchanged. This article covers the 2D formula $\begin{bmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{bmatrix}$, the 3D axis rotations, where the formula comes from, the orthogonal-with-determinant-1 properties, and the sign mistakes that cost marks.\ \ Algebra math Skew Symmetric Matrix - Properties, Theorems, and Examples \ \ A skew symmetric matrix is a square matrix that equals the negative of its transpose, so its diagonal entries are all zero and each entry is the negative of its mirror across the diagonal. This article covers the definition, why the diagonal must be zero, the properties, the symmetric-plus-skew decomposition theorem, the odd-order determinant rule, and six worked examples.\ \ Algebra math Identity Matrix - Definition, Properties, Examples \ \ An identity matrix is a square matrix with 1s along its main diagonal and 0s everywhere else, written I or I_n; multiplying any matrix by it leaves that matrix unchanged. This article covers the definition, the notation, why it acts like the number 1, its determinant and inverse, and six worked examples.\ \ Algebra math Diagonal Matrix - Definition, Properties, Inverse, Examples \ \ A diagonal matrix is a square matrix with zeros off the main diagonal. Learn its properties, determinant, inverse, special cases, and worked examples.\ \ Algebra math Order of a Matrix - Definition, m x n Notation, Examples \ \ The order of a matrix is its size, written as m x n, where m is the number of rows and n is the number of columns. This article covers how to read and find the order, why the order fixes the number of elements, the matrix types defined by order (row, column, square, rectangular), and six worked examples.\ \ Algebra math Matrices and Determinants - Definition, Difference, Examples \ \ Matrices and determinants are two different objects: a matrix is a rectangular grid of numbers, while a determinant is a single number you compute from a square matrix. This article covers both definitions, the exact difference between them, how to find a 2x2 and 3x3 determinant, the properties that make them useful, and six worked examples.\ \ Algebra math Descartes Rule of Signs - Steps & Examples \ \ Descartes rule of signs counts the sign changes in a polynomial's coefficients to bound how many positive and negative real roots it can have — the count of positive roots equals the number of sign changes in $f(x)$, or less by an even number. This article gives the steps, a possible-roots chart, the link to complex roots, six worked examples, and the mistakes that throw off the count.\ \ Algebra math Remainder Theorem - Statement, Proof & Examples \ \ The remainder theorem says that when a polynomial $p(x)$ is divided by $(x - a)$, the remainder is just $p(a)$ — so you find a remainder by substituting, not by dividing. This article states and proves the theorem, shows how it connects to the factor theorem, works six examples, and names the sign and substitution mistakes that cost the most marks.\ \ Algebra math Factorization of Quadratic Polynomials - Methods \ \ Factorizing a quadratic polynomial $ax^2 + bx + c$ means rewriting it as a product of two linear factors, usually by splitting the middle term so the two pieces multiply to $ac$ and add to $b$. This article covers four methods, six worked examples, and the sign errors that derail most attempts.\ \ Algebra math Cube of a Binomial - Formula & Worked Examples \ \ The cube of a binomial expands as $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$, with the middle coefficients 3 and 3 coming from how the three brackets multiply together. This article derives the formula, shows it as a physical cube split into eight blocks, covers the $(a-b)^3$ version, works six examples, and names the term students drop most.\ \ Algebra math Multiplying Binomial - FOIL, Methods & Examples \ \ Multiplying a binomial means distributing every term in the first bracket across every term in the second, so $(a+b)(c+d) = ac + ad + bc + bd$. This article walks through three methods (the distributive property, FOIL, and the box method), six worked examples, and the sign and like-term mistakes that trip students up most.\ \ Algebra math Subtracting Polynomials: Rules, Methods, Examples \ \ Subtracting polynomials means changing the sign of every term in the polynomial being subtracted, then adding like terms as usual. You can work horizontally, distributing the minus sign through the parentheses, or vertically, by flipping signs and adding down columns. This article covers the sign-flip rule, both methods step by step, the single most common sign trap, and six worked examples from two binomials to multivariable polynomials.\ \ Algebra math Adding Polynomials: Rules, Methods, Worked Examples \ \ Adding polynomials means combining like terms: terms with the same variable raised to the same power. You can add them horizontally, grouping like terms in a line, or vertically, stacking them in columns by power. This article covers the like-terms rule, both methods step by step, what happens with unlike terms, the sign and exponent traps that cost marks, and six worked examples from two binomials to multivariable polynomials.\ \ Algebra math Dividing Polynomials: Methods, Synthetic Division, Examples \ \ Dividing polynomials means choosing the right method for the divisor: split terms for a monomial, long division for any larger divisor, and synthetic division as a shortcut when you divide by $x - k$. Every method arranges polynomials in standard form first and inserts zeros for missing powers. This article covers all three methods, how to read the quotient and remainder, when each method applies, and six worked examples from a one-step monomial split to a synthetic division with a remainder.\ \ Algebra math Long Division of Polynomials: Steps, Examples, Remainder \ \ Long division of polynomials follows the same divide, multiply, subtract, bring-down rhythm as long division with numbers, but with variables and powers. Arrange both polynomials from highest power to lowest, insert zeros for any missing terms, then divide leading term by leading term until the remainder's degree drops below the divisor's. This article gives the full method, the missing-term fix, dividing by monomials and binomials, the division algorithm check, and six worked examples.\ \ Algebra math Degree of a Polynomial: Definition, How to Find, Examples \ \ The degree of a polynomial is the highest exponent on its variable once like terms are combined. For one variable, read off the largest power; for several variables, add the exponents in each term and take the largest sum. This article covers how to find the degree step by step, the special cases of constant and zero polynomials, the names of polynomials by degree, six worked examples, and the mistakes that cost marks.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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 $\pm 5$.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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 $\pm 2$.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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).\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math Solve Matrices - Gaussian Elimination, Inverse, Cramer's Rule \ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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).\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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 $\sqrt[3]{64} = 4$.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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: $A^{-1} = \dfrac{1}{\det(A)} \operatorname{adj}(A)$. \ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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 ($\Delta < 0$). \ \ Algebra math 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 ($\log_b 1 = 0$, $\log_b b = 1$, $\log_b b^n = n$).\ \ Algebra math 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 $x = h$.\ \ Algebra math 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 $S_n = \frac{n}{2}(a_1 + a_n)$.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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. \ \ Algebra math 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.\ \ Algebra math 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.\ \ Algebra math 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. \ \ Algebra math 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.\ \ Algebra math 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 $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$).\ \ Algebra math 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 math 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.\ \ Algebra math Perfect Squares - List of Squares and Formula \ \ Algebra math Polynomials – Definition, Types, and Formulas \ \ Algebra