Type an equation, inequality or expression and the calculator works out what to do with it – solve for x, simplify, expand or factor – then explains each step the way a teacher writes it on the board and draws the graph. For example, 2(x + 3) = 5x − 9 gives x = 5. The same box takes systems, trig equations, matrices, sequences and complex numbers.
What it can solve
- Linear equations with brackets, fractions and decimals – fractions are cleared with the LCD first. See the equation solver.
- Quadratic equations by the square-root method, factoring, the quadratic formula or completing the square, including complex roots – quadratic equation solver.
- Polynomial equations of degree 3 and higher: common factors, grouping, substitution such as
u = x², and the rational-root theorem with synthetic division. - Rational equations (the variable in a denominator) with the restrictions stated and extraneous answers rejected.
- Radical, absolute value, exponential and logarithmic equations, each with its own standard method and a check.
- Systems of equations by substitution or elimination, three-variable systems, and simple non-linear systems – system of equations calculator.
- Inequalities – linear, compound, quadratic, rational and absolute value – with a sign chart, interval notation and a number line: inequality calculator.
- Factoring and simplifying: GCF, trinomials, special products, grouping, rational expressions, roots and order of operations – factoring calculator and simplify calculator.
- Literal equations and formulas: solve
2x + 3y = 6fory,A = πr²forr, or a quadratic with letters such asx^2 - (a+b)x + ab = 0. - Functions:
f(x) = x^2 - 4x + 3on its own gives the domain, intercepts, zeros and vertex;f(x) = 2x + 1, f(3)evaluates;f(g(x))composes;inverse of f(x) = (2x + 1)/(x - 3)anddomain of sqrt(x + 3)/(x - 2)work too. - Exponents, roots and logarithms: the laws of exponents (
x^(1/2) * x^(3/2)), simplifying roots (sqrt(12x^3)), rationalising denominators (2/(3 - sqrt(5))), and the laws of logarithms (log(8) + log(2),expand log(x^2 y)). - Complex numbers:
(3 + 2i)(1 - i),(2 + i)/(1 - i),i^2025. - Polynomial long division:
divide x^3 - 2x + 1 by x - 1, and the binomial theorem for powers like(2x + 3)^5. - Substituting values:
2x^2 - 3x + 1 when x = -2. - Trigonometry: equations such as
2sin^2(x) - sin(x) - 1 = 0with the general solution, exact values likecos(5pi/6), and identities (prove tan x + cot x = sec x csc x) – trigonometry calculator. - Matrices and vectors: determinants, inverses, products, row reduction and eigenvalues (
det [[1, 2], [3, 4]]) – matrix calculator; lengths, dot and cross products (angle between <1, 0> and <1, 1>) – vector calculator. - Precalculus: sequences and series (
3, 7, 11, … find the 20th term) – sequence calculator; circles, ellipses, hyperbolas and parabolas – conic sections calculator; partial fractions; polar form and roots of complex numbers.
How the solver chooses a method
A good teacher does not use the same recipe for every problem. Neither does this calculator:
(x − 3)² = 16is solved by taking square roots straight away – expanding first would only make more work.x² − 5x + 6 = 0factors with whole numbers, so it is factored;2x² + 3x − 4 = 0does not, so the quadratic formula is used. The test is whether the discriminantb^2 - 4acis a perfect square.x² = 5xis never divided byx(that losesx = 0) – the common factor is taken out instead.- In a system, a variable with coefficient 1 is isolated (substitution); otherwise the equations are multiplied and added (elimination), choosing the variable that needs the smallest multipliers.
- When the variable appears inside and outside a power or log (
2^x = x + 3), no algebraic method exists, so the answer is found numerically and the solution says so.
How to type maths
| To write | Type | Example |
|---|---|---|
| Power | ^ | x^2, 2^(x+1) |
| Multiplication | nothing, or * | 3x, 2(x+1), 4*2^x |
| Fraction | / with brackets | (x+3)/2, 1/(x-1) |
| Square root, cube root | sqrt( ), cbrt( ) | sqrt(x+3) |
| Absolute value | | | | |2x - 1| |
| Logarithms | log, ln, log2 | log(x) is base 10, log_3(x) |
| Inequality signs | < > <= >= | -3 < 2x + 1 <= 7 |
| Several equations | commas | x + y = 5, x - y = 1 |
| Constants | pi, e | e^(2x) = 7 |
Worked example
Solve 3(x - 2) + 4 = 2x - 5
\begin{aligned} 3x - 6 + 4 &= 2x - 5 && \text{expand the bracket} \\ 3x - 2 &= 2x - 5 && \text{combine like terms} \\ x - 2 &= -5 && \text{subtract } 2x \\ x &= -3 && \text{add } 2 \end{aligned}Check: 3(−3 − 2) + 4 = −11 and 2(−3) − 5 = −11, so x = −3 is correct.
Why answers are checked
Squaring both sides, multiplying by an expression with the variable, or combining logarithms can create extraneous solutions – numbers that come out of the working but do not satisfy the original equation. The calculator puts every candidate back into the equation you typed and rejects the ones that fail, saying why (a negative square root, a zero denominator, the log of a negative number).
Frequently asked questions
Is this algebra calculator free?
Yes. Every answer comes with the full step-by-step solution, with no sign-up and no limit.
What does extraneous solution mean?
A number that comes out of the algebra but does not satisfy the original equation, usually after squaring both sides or multiplying by an expression containing the variable. The calculator checks every answer and rejects these.
Why is my answer given with a square root or as a fraction?
Exact answers are kept exact: 7/2 instead of 3.5 when it repeats, and 3 + √5 instead of 5.2361. The decimal value is shown next to it.
Can it work with functions like f(x)?
Yes. Type f(x) = … on its own for its domain, zeros, y-intercept and (for a quadratic) vertex; add a value like f(3) or f(x + 1) to evaluate it; type “inverse of f(x) = …” or “domain of …” for those.
Why are some steps not numbered?
Conditions you note before solving (like x ≠ 2) and the final check are shown without a number, because they are not part of the working that reaches the answer. Rejecting an extraneous answer is numbered, because it changes the answer.
Can it solve equations with letters other than x?
Yes. Use any letters. With several letters, choose the one to solve for in Solve for, or type “for r” after the equation, like A = pi r^2 for r.