Algebra & equations
Equations, polynomials, factoring and simplification — the solution and the working, not just the answer. 59 tools, including Absolute Value Equation Solver, Absolute Value Inequality Solver, Algebraic Expression Simplifier and Beta Function Calculator. All free, with no account and nothing to install.
Absolute Value Equation SolverSolve absolute-value equations like |x+3| = 5, 3|x+2| = 12 or |x+1| = |x−3| with step-by-step case analysis and exact solutions.Absolute Value Inequality SolverSolve absolute-value inequalities such as |x+3| < 5 or |x−2| > 3 and get the solution as an interval, with AND/OR reasoning explained step by step.Algebraic Expression SimplifierCombines like terms, expands or factors, with implicit multiplication understood (2x is 2·x). When a cancellation removes a factor, the excluded value is stated — (x²−4)/(x−2) is x+2 only for x ≠ 2, and that restriction is part of the answer.Beta Function CalculatorCompute the Euler Beta function B(x, y) = Γ(x)·Γ(y) / Γ(x+y) via a stable log-gamma.Boolean Algebra SimplifierReduces a boolean expression to a minimal sum of products, accepting every common notation — AB, A·B, A&B for AND, A+B or A|B for OR, ~A, !A or A' for NOT, plus XOR, NAND and NOR. Shows the prime implicants, not just the final answer.Characteristic polynomial calculatorEnter a square matrix (2×2 to 5×5) and get its characteristic polynomial p(λ) = det(λI − A), whose roots are the eigenvalues. It uses the Faddeev–LeVerrier algorithm, which is exact at any size, and also reports the trace and determinant.Complementary Error Function CalculatorCompute erfc(x) = 1 − erf(x) to high precision, accurate even far in the tail.Completing the Square CalculatorRewrite ax² + bx + c into vertex form a(x − h)² + k, with the vertex, the completed-square steps and the real or complex roots.Complex Number CalculatorAdd, subtract, multiply and divide complex numbers, with modulus, argument, conjugate and polar form.Cube root calculatorCompute the cube root of a number, plus its cube.Cubic Equation SolverSolve ax³ + bx² + cx + d = 0 for all three roots (real and complex), with the discriminant, the nature of the roots and steps.Descartes' Rule of Signs CalculatorSign changes in f(x) and f(−x) marked one by one, then every possible (positive, negative, complex) combination that adds up to the degree.Digital Root CalculatorCompute the additive and multiplicative digital root of any number in base 2, 8, 10 or 16, with the full chain, the mod-9 shortcut and the persistence count.Domain and Range CalculatorEstimate the domain (excluded x-values) and range of a function f(x) by numerical sampling.Equation of a Line CalculatorFind a line in every common form — slope-intercept, point-slope and standard — at once.Error Function CalculatorEvaluate erf(x), erfc(x) and their inverses erf⁻¹ and erfc⁻¹ to high precision.Expand Polynomials CalculatorExpand products and powers with the working shown — FOIL for two binomials, the binomial theorem for a power, the full distribution grid otherwise.Exponent calculatorRaise a base to any power (exponent).Exponential Decay CalculatorSolve P(t) = P₀·e^(−rt) for any variable, with the half-life.Exponential Equation SolverSolve exponential equations of six types — basic, coefficient, linear exponent, two bases, quadratic-in-a^x and shifted — with logarithms and a full step-by-step solution.Exponential Growth CalculatorSolve P(t) = P₀·(1+r)^t for any variable, with the doubling time.Fraction simplifierReduce a fraction to its simplest form and see its decimal value.Function Composition CalculatorEvaluate and graph (f∘g)(x) and (g∘f)(x) for any two functions, with a value table and point evaluation.Function GrapherPlot up to three functions f, g and h over a custom x- and y-range on an interactive canvas.Function Odd Even Neither CheckerCheck whether a function is even, odd or neither by testing f(−x) against ±f(x).Gamma function calculator Γ(x)Evaluate the gamma function Γ(x), the continuous extension of the factorial with Γ(n) = (n−1)! for positive integers. It uses the Lanczos approximation and Euler's reflection formula to handle negative arguments, and flags the poles at 0, −1, −2, … where Γ blows up. For whole numbers it also shows the exact factorial (x−1)!. Γ appears throughout probability, combinatorics and the analysis of special functions.Hyperbolic functions calculatorAll six hyperbolic functions of a value x at once — sinh, cosh, tanh and their reciprocals coth, sech, csch. Built on the exponential (sinh x = (eˣ − e⁻ˣ)/2), they describe catenary cables, relativity and more. Note coth and csch are undefined at x = 0.Inequality SolverSolve linear, quadratic and rational inequalities such as x^2-5x+6>0 or (x+2)/(x-1)>0 with a full sign chart and the solution set in interval notation.Inverse Function CalculatorSolve f(x) = y numerically for x and see the inverse as the mirror of f across the line y = x.Linear Equation SolverSolve linear equations like 2x + 3 = 7, x/2 + 4 = 7 or 7x − 5 = 3x + 11 in one variable, with exact fractions and step-by-step working.Linear Programming SolverSolve linear programs with the simplex method: maximize or minimize a linear objective under ≤, ≥ and = constraints with up to 8 variables, returning the optimal value, the optimal point and detecting infeasible or unbounded problems.Literal Equation SolverRearrange a formula to solve for any variable — e.g. A = pi*r^2 gives r = sqrt(A/pi) — with genuine symbolic isolation for linear, power and product forms.Log (Logarithm) CalculatorCompute a logarithm in any base — natural, base 10, base 2 or a custom base — using the change-of-base formula.Logarithmic Equation SolverSolve logarithmic equations of six types — basic, linear, two logs, sum of logs, exponential and change of base — with step-by-step working, base e support, and extraneous-root rejection.Möbius function calculator μ(n)The Möbius function μ(n) is 0 if n has a squared prime factor, otherwise +1 or −1 for an even or odd number of distinct prime factors. Enter n and get μ(n) with its prime factorisation and whether n is squarefree.Modular exponentiation calculatorCompute aᵇ mod n efficiently with fast (square-and-multiply) exponentiation and big integers, so even astronomically large powers reduce in a fraction of a second. It normalises a negative or oversized base into the range 0…n−1 first, and supports negative exponents when the base is invertible modulo n. This is the core operation behind RSA, Diffie–Hellman and primality testing.Natural Log CalculatorCompute ln(x), the natural logarithm in base e, to the number of decimal places you choose.Nernst Equation CalculatorCompute a cell potential E = E° − (RT/nF)·ln(Q) at any temperature, from Q or product/reactant concentrations.Nth root calculatorCompute any root (square, cube, nth) of a number.Polar Equation PlotterPlot polar curves r(θ) — roses, cardioids, spirals and more — over any θ range on a polar grid.Polynomial Factoring CalculatorFactor over ℚ with the pattern named at every step: GCF, difference of squares, perfect square, sum or difference of cubes (SOAP), the ac-method and grouping — then verified by re-expansion.Polynomial Long Division CalculatorThe whole tableau, subtraction row by subtraction row, with quotient, remainder and the identity dividend = divisor × quotient + remainder checked for you.Polynomial Roots CalculatorEnter a polynomial equation in x (any degree) and get the degree, all real and complex roots and their multiplicity, with steps.Primitive root calculatorFind the primitive roots modulo n — the generators g whose powers g¹, g², … run through every number coprime to n before repeating (order equal to φ(n)). Primitive roots exist only when n is 1, 2, 4, pᵏ or 2pᵏ for an odd prime p; the tool reports the smallest one, the exact count φ(φ(n)) and lists them. They are the backbone of discrete-logarithm cryptography and Diffie–Hellman key exchange.Quadratic Formula CalculatorSolve ax² + bx + c = 0: get the discriminant, both roots and the nature of the solutions.Quartic Equation SolverSolve ax⁴ + bx³ + cx² + dx + e = 0 for all four roots (real and complex) with a Durand–Kerner numeric solver and step-by-step results.Radical Equation SolverSolve square-root equations by isolating and squaring, then checking every candidate in the original equation to drop extraneous roots — four structured forms with step-by-step working.Radical SimplifierSimplify square, cube and n-th roots, reduce quotients of radicals and rationalize denominators — e.g. sqrt(50) → 5√2 and 1/sqrt(2) → √2/2 — with the exact form and a decimal value.Rational Equation SolverSolve rational (fraction) equations by clearing denominators to a polynomial, then excluding roots that make any denominator zero — reporting solutions, excluded values, no-solution and identity cases.Rational Expression CalculatorSimplify, add, subtract, multiply, divide, factor and decompose rational expressions in x — e.g. (x^2-1)/(x-1) → x+1 — using exact polynomial arithmetic and partial fractions.Rational Root Theorem CalculatorEvery ±p/q candidate, deduplicated and sorted, then actually tested — with the real roots, their multiplicity and the depressed polynomial left after dividing them out.Square root calculatorCompute the square root of a number, plus its square.Synthetic Division CalculatorThe three-row tableau for (x − a), including fractional a such as 1/2 and non-monic divisors, with the remainder theorem and the factor theorem spelled out.System of Inequalities GrapherGraph a system of linear inequalities in x and y: shade the feasible region, draw each boundary (dashed for strict), list the region's vertices and tell you whether it is bounded, unbounded or empty.System of Linear Equations SolverSolve 2×2, 3×3 and 4×4 linear systems by Gaussian elimination, Cramer's rule or the matrix inverse — with the determinant, step-by-step working, and detection of no-solution and infinite-solution cases.System of Nonlinear Equations SolverSolve 2- or 3-variable nonlinear systems (polynomial, trigonometric, exponential) with multivariate Newton's method — enter equations as text, set a starting guess, and surface every real solution with its residual.System of ODEs SolverSolves 2×2 and 3×3 linear systems x′ = Ax with eigenvalues, eigenvectors and the closed-form general solution, classifies the fixed point (node, spiral, centre, saddle, degenerate), and integrates arbitrary nonlinear 2D systems numerically with a phase portrait.Vertex and Axis of Symmetry CalculatorFind the vertex, axis of symmetry and opening direction of any quadratic f(x) = ax² + bx + c.X and Y Intercept CalculatorFind where a curve crosses the x- and y-axes from an equation in x and y (line, parabola, circle…).