Algebra and Calculus Solver: Solve Equations, Inequalities & Systems

Solve equations, inequalities, systems of 2 or 3 variables, trig and logarithmic equations, matrices, and calculus problems — or graph a function and generate random practice problems. Everything runs entirely in your browser — nothing you type is ever sent anywhere.

Equation

Reliable for linear and quadratic equations (quadratics also show the quadratic-formula work). Cubic and quartic equations are solved and then double-checked by substitution, because the solver can occasionally return a wrong root — if that happens you'll see a warning here instead of a silently wrong answer. Use ^ for exponents. Variable is auto-detected (defaults to x).

Inequality

Works via root-finding + sign analysis, so it's solid for polynomial inequalities (linear, quadratic, cubic). It does not correctly handle inequalities with the variable in a denominator (rational expressions) — those need domain restrictions this solver doesn't check. Avoid inputs like 1/(x-2) > 0.

Expression

Shows the simplified, expanded, and factored form. If your expression has no variables, you'll also get a decimal value.

Equation 1

Equation 2

Solves a system of two linear equations in two variables (e.g. x and y), or three equations in three variables (e.g. x, y, z) — click "Add third equation" for the latter.

Equation (variable must be x)

Search domain min

Search domain max

Handles trig equations (sin, cos, tan), logarithms (log = base 10, ln = natural log), and exponentials (2^x = 8). Common forms like log(x) = c or a^x = b get an exact answer; anything else is solved numerically by scanning the domain above for sign changes, so widen the domain if you expect a solution outside it. Trig equations are periodic — solutions repeat outside the domain shown.

Matrix A (one row per line, values comma-separated)

Operation

Matrix B

Enter each row on its own line, values separated by commas (e.g. 1,2 then 3,4 for a 2×2 matrix). Works well up to about 4×4. Determinant, inverse, and solve all require a square matrix A.

Function of x

Uses a separate symbolic-math engine from the equation solver, since it handles derivatives and integrals more reliably. Supports polynomials plus sin, cos, tan, ln, sqrt. Indefinite integrals omit the "+ C" — remember to add it yourself.

Function f(x)

x min

x max

Plots f(x) over the given x-range. Supports sin, cos, tan, log, ln, sqrt, and ^ for powers. Below the graph you'll see the y-intercept and any real x-intercepts found in that range.

Polynomial (in x)

Divide by (x − r) — enter r

Runs synthetic division of a polynomial by a linear factor (x − r) and shows every step, plus the remainder. If the remainder is 0, (x − r) is a factor.

Problem type

Press "New problem" to start

Difficulty: Basic

Your History — Algebra and Calculus Solver

What This Algebra Solver Does

This tool brings together ten of the most common algebra and pre-calculus tasks in one page: solving equations (with full step-by-step work for linear, quadratic, and many cubic equations), solving inequalities, solving systems of two or three linear equations, simplifying expressions, solving trigonometric and logarithmic equations, matrix operations, taking derivatives and integrals, graphing functions, dividing polynomials, and generating random practice problems. It's built for students checking homework, anyone brushing up on algebra, or anyone who just needs a quick answer without opening a full computer algebra system. All of the math happens client-side in your browser, so there's no waiting on a server and nothing you type ever leaves your device.

What Is Algebra?

Algebra is the branch of mathematics that uses letters and symbols, called variables, to represent unknown or changing quantities. Instead of working only with specific numbers, algebra lets you write general rules and relationships — like "twice a number plus five equals seventeen" — and then solve for what that unknown number must be. It's the mathematical language that connects basic arithmetic to more advanced fields like calculus, statistics, and physics, and it underlies nearly every quantitative subject taught after elementary math.

At its core, algebra revolves around a small set of ideas applied over and over: an equation states that two expressions are equal, an inequality states that one expression is greater than, less than, or not equal to another, and an expression is any combination of numbers, variables, and operations that can be simplified but doesn't assert equality or inequality on its own. Learning to move fluently between these forms — solving, simplifying, expanding, and factoring — is what most introductory and intermediate algebra courses are built around.

The Algebra Concepts Behind Each Tool

Equations: An equation is solved by isolating the variable — performing the same operation to both sides until only the variable remains on one side. Linear equations (degree 1) show the isolation steps directly; quadratic equations (degree 2) show the full quadratic-formula work, including the discriminant; many cubic equations (degree 3) show a rational-root-theorem search followed by synthetic division down to a quadratic. Quartic equations still solve, but without a full step breakdown.
Inequalities: Instead of a single value, an inequality's solution is usually a range of values, often written in interval notation like (2, 5] or (-∞, 3). Polynomial inequalities are typically solved by finding the boundary points (where the expression equals zero) and then testing the sign of the expression in each interval between those points.
Systems of Equations: A system of two linear equations in two variables has one solution wherever the two lines intersect on a graph, unless the lines are parallel (no solution) or identical (infinitely many solutions). A system of three equations in three variables works the same way, just in three dimensions — the solution is the single point where all three planes meet.
Expressions: Simplifying combines like terms; expanding multiplies out parentheses (for example, turning (x+3)(x-2) into x² + x - 6); factoring reverses that process, rewriting an expression as a product of simpler terms.
Trigonometric & Logarithmic Equations: Logarithmic and exponential equations of the form log(x) = c or aˣ = b have exact solutions found by applying logarithm rules. Trig equations like sin(x) = 0.5 are periodic — they have infinitely many solutions — so this tool finds every solution within a domain you choose rather than one single answer.
Matrices: A matrix is a rectangular grid of numbers used to represent and solve systems of equations compactly. Multiplying, finding a determinant, or inverting a matrix are the building blocks behind solving larger linear systems and behind topics like transformations in geometry and computer graphics.
Calculus (Derivatives & Integrals): A derivative measures how fast a function is changing at any given point — its instantaneous rate of change. An integral does the reverse: it finds a function whose derivative is the one you started with, and is closely tied to computing area under a curve.
Graphing: Plotting a function visually shows its shape, where it crosses the x-axis (its roots) and y-axis (its y-intercept), and how it behaves as x grows or shrinks — often the fastest way to sanity-check an algebraic answer.
Polynomial (Synthetic) Division: Synthetic division is a shortcut for dividing a polynomial by a linear factor of the form (x − r). If the remainder comes out to zero, that confirms (x − r) is an exact factor of the polynomial — a key step in fully factoring higher-degree polynomials.

Frequently Asked Questions (FAQ)

What kinds of algebra problems can this solver handle?

It solves linear, quadratic, cubic, and quartic equations with step-by-step work, polynomial and quadratic inequalities, systems of two or three linear equations, trigonometric and logarithmic equations, matrix operations, and general algebraic expressions (simplifying, expanding, and factoring). A Calculus tab handles derivatives and integrals, a Graphing tab plots any function of x, and a Division tab performs synthetic division of a polynomial by a linear factor.

Can it solve trigonometric or logarithmic equations?

Yes. Common closed forms like log(x) = c, ln(x) = c, or aˣ = b get an exact algebraic answer. Anything more complex, including trig equations like sin(x) = 0.5, is solved numerically by scanning a domain you specify for sign changes and refining each one with bisection.

Does the algebra solver support matrices?

Yes. The Matrices tab supports addition, subtraction, multiplication, determinant, inverse, transpose, and solving a linear system Ax = b, for matrices entered row by row.

Can I graph a function on this page?

Yes. The Graphing tab plots any function of x over a range you choose, and lists the y-intercept along with any real x-intercepts it can find in that range.

Is this algebra calculator actually free to use?

Yes. Every tool on this page runs entirely in your browser using open-source math libraries, so there's no sign-up, no paywall, and nothing you type is ever sent to a server.

Why does the solver sometimes show a warning about dropped solutions?

For cubic and quartic equations, the underlying symbolic engine can occasionally return a root that doesn't actually satisfy the original equation. This tool substitutes every candidate solution back into the equation and flags or removes any that fail that check, so you never see a silently wrong answer.

What does it mean when an equation has a complex solution?

A complex solution (containing i, the imaginary unit) shows up when a quadratic or higher-degree equation has no real number that satisfies it — for example, x² + 1 = 0 has no real solution because no real number squared equals -1. These solutions are mathematically valid but won't correspond to a point on a standard x-y graph.

Can this tool show its work, or does it just give the final answer?

For quadratic equations it shows the full quadratic-formula work, including the discriminant and what it means. For polynomial division, it shows every step of the synthetic division process. Other tools show the key intermediate forms (simplified, expanded, and factored) rather than a full line-by-line derivation.

How is the random practice mode useful for studying algebra?

The Random Practice tab generates a fresh problem on demand at Basic or Advanced difficulty, lets you type in an answer to self-check, and can reveal the correct answer (with steps, for quadratics) whenever you're stuck. It's built for repeated, low-stakes practice rather than a graded test.

Algebra and Calculus Solver: Equations, Systems, Trig, Matrices & Graphing