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Equation Solver

Step-by-step solutions for linear, quadratic, polynomial, radical, exponential & log equations — inequalities, systems, derivatives, graphing — no signup, works offline

Enter equation, inequality or expression auto-detect
⏎ solve · esc clear
Try an example:
Enter an equation above and click SOLVE — full step-by-step working, a graph and a domain/verification check will appear here.

// How to solve every type of equation

Most free calculators online only handle linear and quadratic equations. This one goes further: type any equation, inequality, or expression below and it is automatically classified and routed to the right method — the quadratic formula, Durand-Kerner root-finding for higher-degree polynomials, logarithm rules, or a numerical bisection scan when nothing else fits. Every result includes full working, a live graph, and a domain or verification check, entirely in your browser.

Linear equations

A linear equation like 3x + 5 = 14 has the variable to the first power only. Isolate it by undoing operations in reverse order: subtract 5 from both sides to get 3x = 9, then divide by 3 to get x = 3. If every x-term cancels out, the equation either has no solution (a contradiction like 0 = 5) or infinite solutions (an identity like 0 = 0).

Quadratic equations

For ax² + bx + c = 0, the quadratic formula gives x = (−b ± √(b² − 4ac)) / 2a. The discriminant Δ = b² − 4ac tells you what to expect before you even solve: Δ > 0 means two distinct real roots, Δ = 0 means one repeated root, and Δ < 0 means two complex roots. This solver also checks for integer factoring whenever it exists, as a second, often faster method.

Cubic, quartic and higher-degree polynomials

Past degree two there is no simple formula — degree 3 and 4 have notoriously unwieldy closed forms, and the Abel–Ruffini theorem proves no general formula exists at all for degree 5 and above. This solver instead uses Durand-Kerner iteration, a numerical method in the same family as Newton-Raphson, to converge on all roots — real and complex — simultaneously. It also checks the rational root theorem first, so a clean root like x = 2 is still shown as an exact value rather than a decimal approximation.

Radical (square root) equations

Solving √(x + 3) = 5 means squaring both sides to get x + 3 = 25, so x = 22. Squaring both sides is the trap: it can introduce extraneous solutions that satisfy the squared equation but not the original. This solver checks every candidate against the domain (the expression under the root must be ≥ 0) and against the original, un-squared equation before accepting it.

Exponential and logarithmic equations

An exponential equation like 2^x = 32 is solved by taking a logarithm of both sides: x = log(32)/log(2) = 5. A logarithmic equation like log(x) + log(x−3) = 1 uses the product rule for logs to become log(x(x−3)) = 1, then converts to exponential form. Note: log( here means base-10, matching how most calculators and textbooks use it — use ln( for the natural logarithm. Both directions require a domain check, since a logarithm's argument must be strictly positive.

Absolute value equations

|2x − 3| = 7 splits into two ordinary equations — 2x − 3 = 7 and 2x − 3 = −7 — because the expression inside the bars could have been positive or negative before the absolute value was applied. If the right-hand side is negative, there's no solution at all, since an absolute value can never be negative.

Inequalities

Inequalities solve like equations with one extra rule: multiplying or dividing by a negative number flips the direction of the inequality. This solver avoids that trap entirely by finding the critical points (where the expression equals zero) and testing the sign in each resulting region, then reporting the solution set in interval notation with a visual number line.

Systems of equations

Two linear equations in two unknowns can be solved by elimination or Cramer's rule — this solver uses the determinant method and also graphs both lines, so you can see the intersection point that the algebra found.

▸ Frequently Asked Questions (24)

Everything below runs the same way it's explained here — no calculator "black box." If your question isn't covered, the about page has a way to reach out.

01. Getting started

What types of equations can this solver handle?
Everything: linear (3x+5=14), quadratic (x²−5x+6=0), cubic and higher-degree polynomials (x³−6x²+11x−6=0), radical (√(x+3)=5), exponential (2^x=32), logarithmic (log(x)+log(x−3)=1), absolute value (|2x−3|=7), linear and quadratic inequalities (2x−3>7), systems of two linear equations, derivatives, and general arithmetic/trigonometric expressions. Anything that doesn't fit a closed-form method falls back to a numerical solver, so you always get an answer.
How do I write multiplication, powers and roots?
Multiplication can be implicit: 2x is read as 2×x. For powers use ^ or the x² button (x^2 or x²). For roots use sqrt(...) or the √ button. Division uses /, and * works for explicit multiplication too.
What's the difference between log( and ln(?
log( is base-10 (log(100) = 2), matching the convention used in most algebra classes and calculators. ln( is the natural logarithm, base e (ln(e) = 1). Mixing these up is one of the most common sources of wrong answers in log equations, so this solver keeps them strictly separate.
How do I enter an inequality or absolute value?
Use the ≤ ≥ < > buttons in the math keyboard (or type them directly) for inequalities, e.g. 2x-3>7. For absolute value, wrap the expression in vertical bars or use abs(...) — both |2x-3|=7 and abs(2x-3)=7 work identically.

02. Linear & quadratic equations

How do I solve a linear equation step by step?
For 3x + 5 = 14: subtract 5 from both sides to get 3x = 9, then divide both sides by 3 to get x = 3. The solver shows every operation performed on both sides and verifies the answer by substituting it back into the original equation.
How does the quadratic formula work?
For ax² + bx + c = 0, the quadratic formula gives the roots as x = (−b ± √(b²−4ac)) / 2a. The term b²−4ac is the discriminant (Δ). If Δ > 0, there are two distinct real roots. If Δ = 0, one repeated root. If Δ < 0, the roots are complex (involve √−1 = i).
What does "no solution" or "infinite solutions" mean?
A linear equation has no solution when it simplifies to a contradiction like 0 = 5 — never true for any x. It has infinite solutions when it simplifies to an identity like 0 = 0 — true for every x (e.g. x + 1 = x + 1).
Does it show factoring as well as the formula?
Yes — whenever a quadratic has integer roots, the solver shows the factored form (like (x−2)(x−3)=0) alongside the quadratic formula method, since factoring is often the faster technique to learn from.

03. Polynomial, radical, exponential & log

How does it solve cubic and quartic equations?
Degree 3 and 4 have closed-form formulas, but they're extremely unwieldy — and degree 5 and above provably has no general formula at all (the Abel–Ruffini theorem). Instead, this solver uses Durand-Kerner iteration, a numerical method that converges on all roots — real and complex — of a polynomial of any degree simultaneously. It's the same family of technique as Newton-Raphson, generalized to find every root at once.
Why do some polynomial roots show as fractions and others as decimals?
The solver checks the rational root theorem first — testing candidates p/q where p divides the constant term and q divides the leading coefficient. Roots that match are shown as exact fractions. Roots that don't correspond to any clean fraction are irrational, so they're shown as decimal approximations, exactly as Wolfram Alpha and other computer algebra tools do.
What are extraneous solutions in radical equations?
Squaring both sides of an equation like √(x+3)=5 can produce solutions that satisfy the squared version but not the original — because squaring erases the sign information the square root depended on. This solver checks every candidate against both the domain (expression under the root ≥ 0) and the original equation, and explicitly lists any rejected extraneous roots.
How are exponential equations like 2^x = 32 solved?
Take a logarithm of both sides: log(2^x) = log(32), then use the power rule to bring the exponent down: x·log(2) = log(32), so x = log(32)/log(2) = 5. The solver substitutes the exponential term algebraically so it can handle more complex exponents too, like 3·2^(x+1)=48.
Why do logarithmic equations need a domain check?
A logarithm is only defined for a positive argument — log(x) requires x > 0. Solving algebraically can produce a candidate that violates this (e.g. a negative x), which must be rejected even though it solved the transformed equation. The solver checks every candidate's domain before accepting it as a final answer.

04. Absolute value & inequalities

How do absolute value equations work?
|2x−3|=7 means the expression inside could be +7 or −7 before the absolute value was applied, so it splits into two cases: 2x−3=7 and 2x−3=−7. If the right-hand side is negative, there's no solution, since an absolute value is never negative.
Why does dividing by a negative number flip an inequality?
Because multiplying or dividing both sides by a negative number reverses their relative order — e.g. 2 < 5, but multiplying by −1 gives −2 > −5, not −2 < −5. This solver sidesteps the rule entirely by finding critical points and testing the sign of the expression in each region, so the direction is always correct without manual flipping.
What is interval notation?
It's a compact way to write a solution set: (2, 5] means all x with 2 < x ≤ 5 — a round bracket excludes the endpoint, a square bracket includes it. (−∞, 3) means everything less than 3. The number-line graph below each inequality result shows the same thing visually, with open or filled circles at the endpoints.

05. Systems, derivatives & graphing

How many equations can the system solver handle?
Currently two linear equations in two unknowns (x and y), separated by a semicolon (e.g. 2x+y=5; x-y=1). It solves via the determinant (Cramer's rule) method and plots both lines so you can see the intersection point that the algebra finds — or confirm visually why parallel lines have no solution.
What differentiation rules does the derivative solver use?
Power rule, product rule, quotient rule, and chain rule, applied symbolically via math.js — the same open-source library used for all of the algebra. It handles polynomials, trig functions, exponentials and logarithms.
What does the graph under each result actually show?
For an equation, it plots y = (left side) − (right side) — the solutions are exactly where this curve crosses the x-axis, marked with a dot. For a system, it plots both lines with their intersection marked. For an inequality, a number line shows the solution region shaded, with open circles for strict inequalities and filled circles for ≤/≥.
What happens with trigonometric or other equations it can't solve symbolically?
It falls back to a numerical solver: scanning x from −50 to 50 and refining every sign change with bisection. This works for essentially any equation, including trig and mixed forms, but only finds solutions within that range — periodic equations like sin(x)=0.5 have infinitely many solutions that repeat beyond it, which the result note points out.

06. Accuracy, privacy & technical

How accurate is this compared to other equation solvers?
The algebraic methods (linear, quadratic, radical, exponential, log, absolute value) are exact closed-form solutions — not approximations. Higher-degree polynomial roots use Durand-Kerner iteration, which converges to machine precision (differences below 1e-10) for well-conditioned polynomials, the same numerical standard used by professional computer algebra systems.
Is my calculation private? Does it get sent to a server?
Everything runs 100% in your browser. No calculation, equation, or result is ever sent to a server. The solver uses math.js, an open-source mathematics library loaded once and then running entirely client-side. Your history is stored locally in your browser only.
Can I share my equation with someone?
Yes — click share to copy a link to your clipboard. The equation is encoded in the URL, so anyone with the link opens the solver with the same problem pre-filled and solved.
Does it work offline or without an account?
Yes to both. There's no signup, no login, and no ads. Once the page has loaded once, the math engine runs entirely client-side, so it keeps working without an internet connection.