Math & Statistics
Algebra Calculators
Calculators for solving equations, rewriting expressions, and working with matrices and sequences. The Quadratic Formula Calculator returns both roots together with the discriminant that decides whether they are real, and the Cubic Equation Calculator solves third-degree equations exactly. The System of Equations Calculator solves simultaneous linear equations; the Factoring Trinomials Calculator and the Polynomial Division Calculator rewrite a polynomial as factors or as a quotient and remainder, and the Partial Fraction Decomposition Calculator splits a rational expression into simpler fractions. For linear algebra, the Matrix Calculator does matrix arithmetic, while the Determinant Calculator and the Eigenvalue Calculator give the numbers that decide whether a system has one solution and how a transformation scales space. The Arithmetic Sequence Calculator and the Geometric Sequence Calculator find any term and partial sums, the Log Calculator and the Logarithm Rules Calculator evaluate and rewrite logarithms, and the Exponential Growth Calculator models growth and decay over time.
Algebra
- Quadratic Formula CalculatorSolve any quadratic with exact roots — surds stay surds and a negative discriminant gives the complex pair — plus the vertex, the factored form and every step of the working.
- Descartes' Rule of Signs CalculatorCount the sign variations in f(x) and f(-x), list every combination of positive, negative and complex roots the rule permits, and see which one is actually true.
- Inequality to Interval Notation CalculatorSolve a linear, chained or absolute-value inequality and write the answer in interval notation, with the sign flip marked at the step where it happens and exact fractional endpoints.
- Discriminant CalculatorEvaluate b² − 4ac, the cubic and quartic discriminants, or D = f_xx·f_yy − (f_xy)² for the second-derivative test, in exact arithmetic so the sign is never a rounding artefact.
- Completing the Square CalculatorRewrite ax² + bx + c as a(x − h)² + k with every step shown, solve by the method, or find the constant that completes a square — all in exact fractions.
- Factoring Trinomials CalculatorFactor a trinomial by the AC method with every factor pair of ac listed, the common factor taken out first, and the answer multiplied back out to check it.
- System of Equations CalculatorSolve linear systems in up to six unknowns with exact fractions — row operations, the matrix form, Cramer’s rule, and the parametric family when there is no single answer.
- Elimination Method CalculatorSolve simultaneous equations by elimination with the working shown, including which unknown to remove first and the whole-number multipliers that keep it tidy.
- Substitution Method CalculatorSolve simultaneous equations by substitution, with the unknown to isolate chosen and explained, the bracketed substitution as its own line, and the expansion after it.
- FOIL CalculatorMultiply two binomials with all four products named — First, Outer, Inner, Last — shown as a grid too, with difference of squares and perfect squares recognised.
- Multiplying Polynomials CalculatorMultiply polynomials of any size in any number of variables, by grid, vertical or term-by-term layout, with the product count that catches a dropped term.
- Adding and Subtracting Polynomials CalculatorAdd or subtract polynomials with the sign flip shown as its own step, like terms aligned in columns, and a visible gap wherever a power is missing.
- Polynomial Division CalculatorDivide polynomials by long division or synthetic division, with the quotient, the remainder, the remainder theorem check and every stage of the working.
- Cubic Equation CalculatorSolve any cubic exactly when it has a rational root — deflate and finish with the quadratic formula — and by the trigonometric form when it does not, with the discriminant saying which case you are in.
- Rational Zeros CalculatorList every rational root candidate from the rational root theorem, test each one, divide out the hits by synthetic division, and use the bound rules to strike the rest off without testing them.
- Perfect Square Trinomial CalculatorTest whether a trinomial is a perfect square against all three conditions, name which one failed, and give the constant that would fix it — plus the difference-of-squares answer when that is what you actually have.
- Square of a Binomial CalculatorExpand (a plus or minus b) squared with the middle term named and a numeric example of exactly what the (a+b)^2 = a^2+b^2 mistake costs, plus the area model that makes it stick.
- Diamond Problem CalculatorFill in a diamond from any two of its four cells, including the product-and-sum case that is really a quadratic — with an impossible diamond reported as impossible rather than rounded.
- Direct Variation CalculatorFind the constant of variation from one pair and predict any other, for y = kx or y = kx^n — and test whether a table really is a direct variation or just a straight line that misses the origin.
- Inverse Variation CalculatorFind k from one pair and predict any other for y = k/x or y = k/x^n, with the asymptotes treated as content, plus joint and combined variation in the same place.
- Interval Notation CalculatorConvert between inequality, interval notation, set-builder form and a number-line graph in every direction, including absolute values, unions and the empty set.
- Graphing Quadratic Inequalities CalculatorSolve a quadratic inequality by sign chart rather than by four memorised cases, and shade the region for the two-variable form, with the repeated-root case where all four relations differ.
- Multiplying Binomials CalculatorMultiply two to six binomials with the 2^n product count as the check, the build-up shown one bracket at a time, and the roots reading that turns a product of (x − r) into its polynomial.
- Graphing Inequalities on a Number LinePlot up to six inequalities on one shared number line, read the overlap and the union straight off it, and see which conditions conflict and which are doing no work at all.
- Absolute Value Equation CalculatorBoth cases written out, in interval and set-builder notation — with the distinction the topic turns on: a less-than is an AND, a greater-than is an OR.
- Logarithm Rules CalculatorExpand or condense a logarithm step by step, with the rule that produced each term — and an explicit statement that log(a + b) is not log a + log b.
- Partial Fraction Decomposition CalculatorSplit a rational function into simple fractions by the cover-up method: one substitution per term, with improper fractions and repeated roots properly refused.
Powers & Roots
- Log CalculatorLogarithms in any base with the exponential form beside them, whole answers confirmed by raising the base back, and change of base worked through.
- Simplify Radicals CalculatorSimplest radical form at any index from 2 to 12 with a coefficient in front, shown prime by prime, plus the rationalised denominator.
Growth
- Doubling Time CalculatorHow long a quantity takes to double at a constant rate, or from two measurements — with the rule of 70 and rule of 72 measured against the exact answer.
Sets & Logic
- Power Set CalculatorList all 2^n subsets with the binary word that generates each one, grouped by size and checked against the binomial coefficients, with the element-versus-subset distinction spelled out.
- Subset CalculatorTest whether one set is a subset, a proper subset, equal or disjoint — and when the answer is no, name the exact element that breaks it.
- Union and Intersection CalculatorSix set operations on two or three sets at once, with the Venn regions, inclusion-exclusion checked against the real counts, and De Morgan verified on your own sets.
- Truth Table GeneratorBuild a truth table with a column for every sub-expression, classify the result, compare two expressions with a named counterexample row, and read DNF and CNF straight off the table.
Complex Numbers
- Complex Number CalculatorAdd, subtract, multiply, divide and raise complex numbers to powers exactly — (1+i)^8 is 16, not 15.999999999999996 — with the conjugate trick shown as the working for division.
- Complex Conjugate CalculatorThe conjugate, and everything it does: the sum, difference, product and quotient identities computed on your number, the reflection geometry, and the real quadratic behind conjugate root pairs.
- Complex Root CalculatorAll n roots of any complex number by De Moivre, exact wherever a root is a Gaussian rational, arranged as the regular n-gon they form, with the sum and product checked.
- Quaternion CalculatorExact quaternion arithmetic showing both pq and qp every time, both quotients rather than one, and the rotation a unit quaternion represents with axis, angle, matrix and the gimbal-lock case named.
Special Functions
- Bessel Function CalculatorJ, Y, I and K plus the spherical pair, with the zeros that set drum modes and waveguide cutoffs, the recurrence and Wronskian checked, and the measured accuracy stated per family.
Sequences
- Arithmetic Sequence CalculatorNth term and sum with both formulas substituted, carried as exact fractions — a step of 0.1 gives exactly 4 at term 40 rather than 3.9999999999999996.
- Geometric Sequence CalculatorNth term, partial sum, and whether the infinite series converges at all — with exact ratios, so 1/3 stays 1/3 instead of becoming 0.3333333333.
- Fibonacci CalculatorEvery digit of F(n), not a rounded double — a JavaScript number stops being exact at F(79), and F(80) is where most web calculators quietly go wrong.
- Pascal’s Triangle CalculatorThe triangle drawn to any row up to 40, exact on big integers, with the row sum as a power of two and each diagonal's meaning named.
- Collatz Conjecture CalculatorHalve it or triple it until 1 — with the climb drawn on a log scale, the peak, the stopping time, and a plain statement that the conjecture is unproven.
- Nth Term CalculatorGive it the terms and it finds the rule — arithmetic, geometric, quadratic or Fibonacci-like — with the difference table, and an honest 'no rule found' when there is none.
Growth & Decay
- Exponential Growth CalculatorGrowth, decay, doubling time and half-life from one rate — with the widening period-on-period change that is what exponential actually means.
Linear Algebra
- Matrix CalculatorAdd, multiply, transpose and power matrices with exact fractions — showing both AB and BA, because those are different matrices and often not even the same shape.
- Determinant CalculatorThe determinant of any square matrix up to 8×8, exact — by fraction-free elimination rather than the cofactor expansion that becomes unusable past 4×4.
- Matrix Inverse CalculatorThe inverse in exact fractions, with the product A × A⁻¹ displayed — and it is exactly the identity, not the identity plus rounding noise.
- Row Echelon Form CalculatorReduced row echelon form with every step shown, plus rank, nullity, pivot columns and a null space basis — all exact, because rank is a question about zeros.
- Eigenvalue CalculatorEigenvalues and the characteristic polynomial up to 6×6 — the polynomial exact and the roots decimal, because roots of a polynomial are generally irrational.
- Matrix Decomposition CalculatorLU, QR and Cholesky — with LU exact and the other two decimal, because square roots are irrational, and the page labels which is which rather than presenting them alike.
Compare algebra calculators
What each calculator is best for, the main inputs it needs, what it shows, and what to keep in mind.
| Calculator | Best for | Key inputs | Shows | Keep in mind |
|---|---|---|---|---|
| Doubling Time Calculator | How long a quantity takes to double at a steady rate | A growth rate per period, or two measurements and the time between them | The exact doubling time against the rule of 70 and rule of 72 | A falling quantity has a halving time and no doubling time at all |
| Quadratic Formula Calculator | Solving any quadratic, factorable or not | a, b and c, or a typed equation with terms on either side of the equals sign | Exact roots against their decimals, with the discriminant, vertex and factored form | Minus b when b is already negative, and squaring a negative: (-7)squared is +49 |
| Discriminant Calculator | What the roots are like before you find them | Coefficients of a degree 2, 3 or 4 polynomial, or three second partials | The sign, and for a quadratic whether it is also a perfect square | D greater than zero alone does not mean a minimum; the sign of f_xx decides which |
| Completing the Square Calculator | Vertex form, and the method an exam question names | a, b and c, in vertex-form, solve, or find-the-constant mode | The vertex form against its own expansion, coefficient by coefficient | Add AND subtract the square; adding alone changes the expression |
| Factoring Trinomials Calculator | Factoring over the integers, with the AC search shown | Any trinomial, two-term square or cube, or four terms to group | Every factor pair of ac against its sum, and the answer against its expansion | Take the common factor out first; factors that are still divisible are marked wrong |
| System of Equations Calculator | Linear systems of any size, and why one fails | One equation per line, up to six unknowns, terms on either side | The solution against the rank, the reduced matrix and Cramer determinants | A zero determinant does not say whether there are none or infinitely many |
| Elimination Method Calculator | Simultaneous equations by the named method | Two or three equations, in any arrangement | The chosen unknown against the cost of the alternative | Equal coefficients cancel on subtraction, opposite ones on addition |
| Substitution Method Calculator | Systems where one unknown has a coefficient of one | Two equations, including one already solved for a variable | The isolated expression against its bracketed substitution | A minus in front of the brackets reaches every term inside them |
| FOIL Calculator | Multiplying exactly two binomials | Two terms in each bracket, in one variable or two | All four named products against the collected answer | FOIL names four products; three terms in a bracket gives six and is refused |
| Multiplying Polynomials Calculator | Any product bigger than two binomials | Two to five factors, any number of terms and variables | Grid, vertical and listed layouts of the same products | Count the products first; nothing later reveals a dropped one |
| Adding and Subtracting Polynomials Calculator | Combining polynomials, especially subtracting one | Two to five polynomials, any number of variables | Columns of like terms against the one-line version | Subtracting flips every sign in the bracket, not just the first |
| Polynomial Division Calculator | Dividing one polynomial by another | A dividend and a divisor of any degree | Long division against synthetic, and the remainder against P(c) | Synthetic division only works for a linear divisor |
| Multiplying Binomials Calculator | Products of two to six binomials, and roots to polynomial | Two to six bracketed binomials, any variables | The 2ⁿ product count against the terms that survive collecting | A bracket with three terms breaks the 2ⁿ count and is refused |
| Perfect Square Trinomial Calculator | Testing whether a trinomial is a perfect square, and why not | Any quadratic trinomial, fractional coefficients included | All three conditions separately, with the failing one named | Two square end terms do not make a square trinomial; the middle decides |
| Square of a Binomial Calculator | Expanding (a ± b)² with the middle term made unmissable | Two terms and a sign, in any number of variables | The expansion against the a² + b² mistake, priced as a number | (a + b)² is not a² + b²; it is short by exactly 2ab |
| Diamond Problem Calculator | Filling a diamond from any two of its four cells | Any two of product, sum, and the two numbers | The discriminant against whether a real pair exists at all | S² − 4P negative means no such pair exists, not that it is hard to spot |
| Cubic Equation Calculator | Solving a cubic, exactly where a rational root exists | a, b, c and d, or the cubic written out | The discriminant against the root pattern, and Vieta against the roots | Three real roots with no rational one needs the trigonometric form, not Cardano |
| Rational Zeros Calculator | Every rational root, with the whole search shown | Any polynomial of degree 2 to 8 | Each ±p/q candidate against the value of f there | A zero constant term makes the candidate list infinite; factor x out first |
| Descartes' Rule of Signs Calculator | How many roots of each sign, before solving | Any polynomial of degree 1 to 12 | Every combination the rule permits against the true split | Zero coefficients are skipped, not counted as sign changes |
| Direct Variation Calculator | Proportional relationships, y = kx or y = kxⁿ | One known pair, a power, and a value to predict from | Every ratio in a table against the others | A straight line that misses the origin is linear but not proportional |
| Inverse Variation Calculator | Inverse, joint and combined variation | One known pair, a power, or several varying quantities | Every product in a table against the others | x = 0 is not in the domain at all; it is an asymptote, not a value |
| Interval Notation Calculator | Converting between the four ways of writing a solution set | An inequality, an interval, set-builder, or an absolute value | All four written forms against each other and against a number line | Infinity always takes a round bracket; it is not a number any set contains |
| Inequality to Interval Notation Calculator | Solving an inequality, then writing it as an interval | A linear, chained, or absolute-value inequality | The solved answer against substitution of a value inside and outside | Dividing by a negative reverses the inequality; nothing else does |
| Graphing Inequalities on a Number Line | Comparing several conditions on one shared scale | Up to six conditions in any written form | Each condition against the AND and OR rows | A condition containing another entirely is doing no work at all |
| Graphing Quadratic Inequalities Calculator | Quadratic inequalities, on a line or as a shaded region | A quadratic inequality, in one variable or two | Each region's sign against a test value inside it | A negative leading coefficient reverses which side satisfies the inequality |
| Power Set Calculator | Listing every subset, and why there are 2ⁿ | A finite set of up to twelve elements | The count of each size against the binomial coefficients | ∅ and the set itself are both subsets, so P(∅) has one element, not none |
| Subset Calculator | Deciding containment, and naming what breaks it | Two finite sets | Every element of A against membership of B | ∈ relates an element to a set; ⊆ relates two sets, and they are rarely both true |
| Union and Intersection Calculator | Six set operations at once, with the identities checked | Two or three sets, plus a universe for complements | Inclusion–exclusion and De Morgan, both computed rather than quoted | A complement has no meaning until the universal set is stated |
| Truth Table Generator | Truth tables with a column for every sub-expression | A propositional expression in up to six variables | Two expressions row by row, with the first disagreement named | T and F are read as variables here; write TRUE or FALSE for the constants |
| Complex Number Calculator | Exact complex arithmetic, including division | Two complex numbers, or one and a whole-number power | The exact rectangular answer against its polar approximation | Polar form is a decimal; the rectangular result is the exact one |
| Complex Conjugate Calculator | What the conjugate does, not just how to write it | One complex number | |z| against |z̄|, and the five identities against their values | Conjugate root pairs need REAL polynomial coefficients, or the pairing fails |
| Complex Root Calculator | All n roots of a complex number, not only the principal | A complex number and a root from 1 to 24 | Each root raised back to the nth power against the original | Zero has one nth root, not n; the circle they sit on collapses |
| Quaternion Calculator | Quaternion arithmetic and the rotation it represents | Two quaternions, or one read as a rotation | pq against qp, and both quotients against each other | ij = k but ji = −k, so p/q is ambiguous until you say which side |
| Bessel Function Calculator | J, Y, I and K, and the zeros physical problems need | A kind, an integer order 0 to 30, and a value | The recurrence and the Wronskian at your own argument | The zeros are not evenly spaced, which is why a drum has no pitch |
Best calculator path
- Use the Doubling Time Calculator to turn a growth rate into the time a quantity takes to double — exactly, rather than by dividing 72 by it.
- Use the Quadratic Formula Calculator to solve any quadratic exactly, with the substitution written out before it is evaluated and the surd kept as a surd.
- Use the Discriminant Calculator to find out what the roots will be like before finding them, for a quadratic, a cubic, a quartic, or a critical point of a surface.
- Use the Completing the Square Calculator for the vertex form a(x − h)² + k, for solving by that method, or for the constant that completes a square.
- Use the Factoring Trinomials Calculator to factor over the integers by the AC method, with every factor pair of ac shown beside its sum.
- Use the System of Equations Calculator for two or more linear equations at once, in up to six unknowns, with the rank that says why a system has no single answer.
- Use the Elimination Method Calculator when the question names that method, and see which unknown is cheaper to remove.
- Use the Substitution Method Calculator when one unknown has a coefficient of one, or when an equation is already written as y = something.
- Use the FOIL Calculator for exactly two binomials, with all four products named — and the Multiplying Polynomials Calculator for anything larger.
- Use the Multiplying Polynomials Calculator for products of any size, with the grid, vertical and listed layouts and the product count that catches a missing term.
- Use the Adding and Subtracting Polynomials Calculator to combine polynomials, with the sign flip shown before anything is added.
- Use the Polynomial Division Calculator for long or synthetic division, with the remainder theorem checked against the division itself.
- Use the Multiplying Binomials Calculator for three or more binomials, where FOIL runs out of letters but the 2ⁿ product count does not — and to turn a set of roots into the polynomial that has them.
- Use the Cubic Equation Calculator for degree 3, with the discriminant naming the root pattern before anything is solved.
- Use the Rational Zeros Calculator to run the p/q search in full, including the bound rules that strike most candidates off untested.
- Use Descartes’ Rule of Signs to bound how many positive and negative roots there can be, before finding any of them.
- Use the Square of a Binomial Calculator to expand (a ± b)² with the middle term priced, and the Perfect Square Trinomial Calculator to recognise one going the other way.
- Use the Diamond Problem Calculator for the two numbers with a given product and sum — the AC step of factoring by grouping.
- Use the Direct Variation Calculator when a ratio stays fixed, and the Inverse Variation Calculator when a product does.
- Use the Inequality to Interval Notation Calculator to solve an inequality with the sign flip marked where it happens, then the Interval Notation Calculator to write it any of four ways.
- Use the number-line page to compare several conditions on one shared scale and see which of them are doing no work.
- Use the Graphing Quadratic Inequalities Calculator for a quadratic one, by sign chart rather than by four memorised cases.
- Use the Power Set, Subset and Union and Intersection calculators for finite sets, and the Truth Table Generator for the same algebra in logical notation.
- Use the Complex Number Calculator for exact complex arithmetic, the Complex Root Calculator for all n roots, and the Quaternion Calculator where multiplication stops commuting.
- Use the error, gamma and Bessel function calculators for the three special-function families here, each with its measured accuracy stated per family.
- When the math feeds a real decision, carry it into a related tool such as the Compound Interest Calculator to see the money effect over time.