Algebra Calculator
Solve linear and quadratic equations in x with the working shown, simplify expressions, or evaluate them at any value of x — exactly, in fractions and roots, not just rounded decimals.
What This Calculator Can and Cannot Do
It can: solve linear and quadratic equations in the single variable x, with brackets, fractions and negative coefficients; report roots exactly — as fractions, in √ (surd) form, or as complex numbers when there are no real roots; simplify polynomial expressions up to degree 6; and evaluate any such expression at a value of x you choose. Every solve shows its working and finishes by substituting the answer back into your original equation.
It cannot: solve systems of equations (two variables at once), inequalities, equations with x under a root or in a denominator (like √(x) = 4 or 1/x = 2), trigonometric, logarithmic or exponential equations, or equations of degree 3 and above. Type one of those and the calculator will tell you plainly that it is out of scope — it will never guess or hand back a plausible-looking wrong answer. For a different kind of unknown — the missing side of a right triangle — the Pythagorean Theorem Calculator solves a² + b² = c² instead.
Worked Examples (Each One Verified Against the Solver)
Linear. 3(x − 2) = 2x + 5. Expand the bracket: 3x − 6 = 2x + 5. Move the x terms together and the numbers together: x = 5 + 6 = 11. Check: 3 × (11 − 2) = 27 and 2 × 11 + 5 = 27 — both sides agree.
Quadratic. x² − 5x + 6 = 0. The discriminant is D = (−5)² − 4 × 1 × 6 = 25 − 24 = 1, a perfect square, so the roots are rational: x = (5 ± 1) / 2, giving x = 3 or x = 2. Check: 9 − 15 + 6 = 0 and 4 − 10 + 6 = 0.
Simplify. 2(x + 3) + 4x. Expand the bracket: 2x + 6 + 4x. Collect like terms: 6x + 6.
Evaluate. x² + 3x − 4 at x = 2: 4 + 6 − 4 = 6.
How the Solver Works
The page carries a complete symbolic mathematics engine (the open-source nerdamer library, bundled into the page — no server, no uploads). Your input is first expanded into a standard polynomial using exact fraction arithmetic, so 0.1 is treated as 1/10, not as a rounded decimal. Quadratics are then scaled to whole-number coefficients and solved through the discriminant, which is what decides the shape of the answer: a perfect-square discriminant gives rational roots, any other positive value gives exact √ roots, zero gives a repeated root, and a negative value gives complex roots. Because the arithmetic is exact, the check at the end is a real check — the roots go back into your original equation and both sides must agree.
Algebra Calculator — Facts at a Glance
- Answers are exact first — fractions, √ forms and complex roots — with decimals given as approximations, never instead of the exact value.
- The discriminant D = b² − 4ac decides everything about a quadratic: positive means two real roots, zero means one repeated root, negative means two complex roots.
- Every solve is verified on the page by substituting the roots back into the original equation.
- The whole calculation runs locally in your browser; nothing you type is sent to a server.
- Notation it understands: 2x for 2 × x, x^2 or x² for x squared, brackets, fractions like 3/4, and decimals.
Frequently Asked Questions
What kinds of equations can this algebra calculator solve?
Equations in one variable, x, that are linear (like 2x + 3 = 7) or quadratic (like x² − 5x + 6 = 0), including brackets and fractions — everything is first expanded into standard form, then solved exactly. It also simplifies polynomial expressions and evaluates them at a chosen value of x. It does not solve systems of equations, inequalities, or equations involving roots, logarithms, trigonometry or x in a denominator; the limits are listed in full on this page, and anything outside them gets an honest message rather than a wrong answer.
Does the algebra calculator show its working, step by step?
Yes. For a linear equation you see the terms moved to one side, the constant moved across, and the final division. For a quadratic you see the standard form, the discriminant D = b² − 4ac worked out in numbers, and how the roots follow from it — rational roots as exact fractions, irrational roots in exact √ form with a decimal, and complex roots when D is negative. Every solve ends with a check line: the roots are substituted back into your original equation and both sides are shown to agree.
What if my quadratic equation has no real solutions?
Then the discriminant is negative and the calculator says so, showing the complex roots instead of pretending there are none. The simplest case is x² + 1 = 0, whose solutions are x = i and x = −i. Complex roots are not an error — they are the complete answer, and the check line verifies them in complex arithmetic the same way real roots are verified in ordinary arithmetic.
Is the algebra calculator free — and is my input private?
Free, with no signup, and private by construction: the solver is a mathematics library bundled into this page and running entirely in your browser. Your equations are never typed into a server, sent anywhere, or stored — you can confirm it in your browser's network tab, which shows no request leaving the page when you solve. That also means it works offline once the page has loaded.
What is the difference between solving, simplifying and evaluating?
Solving finds the value of x that makes an equation true (3(x − 2) = 2x + 5 gives x = 11). Simplifying rewrites an expression in its tidiest equivalent form without finding x (2(x + 3) + 4x becomes 6x + 6 — true for every x). Evaluating substitutes one chosen value and computes a number (x² + 3x − 4 at x = 2 gives 6). The three modes above correspond to those three jobs.