AnyTool
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How do I solve an equation step by step?

Choose the equation type in AnyTool Equation Solver and type the coefficients. A linear equation ax + b = 0 gives x = −b/a; a quadratic ax² + bx + c = 0 is solved by the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, where the discriminant b² − 4ac decides whether the roots are two real, one repeated, or a complex conjugate pair a ± bi; cubics are solved with Cardano’s method; and 2×2 or 3×3 systems are solved by Cramer’s rule using determinants. Every answer is shown with worked steps, live in your browser.

  • Linear ax + b = 0 → x = −b/a (handles a = 0: no solution or all reals)
  • Quadratic: discriminant b² − 4ac → real, repeated, or complex a ± bi roots
  • Quadratic also gives the vertex (−b/2a, …) and axis of symmetry
  • Cubic by Cardano; 2×2 and 3×3 systems by Cramer’s rule (determinants)
  • 100% in your browser — no upload, no signup, works offline

What is

Equation Solver

An equation solver finds the values of the unknown that make an equation true. For polynomials this means the roots — a linear equation has one, a quadratic up to two, a cubic up to three — while a system of linear equations has a single point (or none, or infinitely many) where all the equations hold at once. The discriminant of a quadratic, b² − 4ac, tells you whether its roots are real and distinct, real and repeated, or a complex conjugate pair.

Calculators

Related terms

Quadratic formulaDiscriminantComplex rootsCramer’s ruleCubic equation

Frequently Asked Questions

Use x = (−b ± √(b² − 4ac)) / 2a. The sign of the discriminant b² − 4ac tells you whether the roots are real, repeated, or complex.

For ax² + bx + c = 0 the quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. The discriminant Δ = b² − 4ac decides the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives a pair of complex conjugate roots a ± bi. AnyTool Equation Solver computes Δ, substitutes into the formula, and also reports the vertex (−b/2a, f(−b/2a)) and the axis of symmetry — all with the steps shown.

It means the discriminant is negative, so there is no real solution; the two roots are complex conjugates written as a + bi and a − bi.

A quadratic has complex roots when its discriminant b² − 4ac is negative, because the formula then asks for the square root of a negative number. The two solutions are complex conjugates, a + bi and a − bi, where the real part is −b/2a and the imaginary part is √(4ac − b²)/2a. They are genuine solutions over the complex numbers, just not on the real number line, so AnyTool shows them as a ± bi rather than reporting an error.

Use Cramer’s rule: divide the determinant with each column replaced by the constants by the coefficient determinant. If the coefficient determinant is zero there is no unique solution.

A 2×2 or 3×3 system is solved by Cramer’s rule. Compute the coefficient determinant D; then for each variable replace that variable’s column with the constants and compute a new determinant (Dx, Dy, Dz), and the value is that determinant divided by D. If D = 0 there is no unique solution: the system is either inconsistent (no solution) or dependent (infinitely many), and AnyTool checks which and says so instead of dividing by zero.

Yes, it is free and runs entirely in your browser. It gives numeric solutions with steps but is not a symbolic computer-algebra system.

AnyTool Equation Solver is completely free with no signup, and every calculation runs in your browser on a small, unit-tested engine, so nothing is uploaded. It returns numeric solutions — real, repeated and complex roots, plus system solutions — each with step-by-step working. It is deliberately not a full computer-algebra system: it does not return symbolic forms like (3 ± √5)/2, factor polynomials, or simplify radicals, and results use IEEE-754 double precision.

Detailed Explanation

Methodology

How the Equation Solver Finds Roots

AnyTool Equation Solver computes solutions in closed form entirely in the browser. A linear equation ax + b = 0 is solved as x = −b/a; a quadratic ax² + bx + c = 0 is solved with the quadratic formula after computing the discriminant b² − 4ac, which decides whether the roots are two real, one repeated, or a complex-conjugate pair; a cubic ax³ + bx² + cx + d = 0 is solved with Cardano’s method (depressed-cubic substitution and the trigonometric form for three real roots); and systems of linear equations are solved with Cramer’s rule using determinants. Every result is produced by direct algebra, not by guessing, and the working is shown step by step.

  • Linear via x = −b/a, with degenerate cases detected
  • Quadratic via the discriminant and the quadratic formula
  • Cubic via Cardano’s method, including the three-real-root case
  • Linear systems (2×2 and 3×3) via Cramer’s rule and determinants
  • Complex roots returned as a + bi when the discriminant is negative
How It Works

Modes, Steps and Complex Roots

Five modes are offered — linear, quadratic, cubic, a 2×2 system and a 3×3 system — each with its own coefficient fields. The solver returns the exact root or roots, shows complex-conjugate solutions as a + bi when a discriminant is negative, and for quadratics it also reports the vertex and axis of symmetry. The standout feature is the step-by-step working: it prints the discriminant, the substituted formula and the simplified result so the answer can be checked and learned from rather than merely copied.

  • Linear, quadratic, cubic, 2×2 and 3×3 system modes
  • Step-by-step working shown for every solution
  • Complex roots displayed as a + bi
  • Quadratic vertex and axis of symmetry included
  • Degenerate cases (no solution / infinitely many) reported clearly
Limitations

What It Is Not

The solver works on polynomial equations up to degree three and on linear systems of two or three unknowns; it is not a full computer-algebra system, so it does not factor symbolically, simplify expressions, handle higher-degree or transcendental equations (such as sin x = x), or solve inequalities. Results are computed with IEEE-754 double precision, so very large coefficients or ill-conditioned systems can carry small floating-point rounding, and Cardano’s trigonometric branch is used to keep three-real-root cubics accurate. These limits are stated plainly so the tool is used where it is exact.

  • Polynomials up to cubic and 2×2 / 3×3 linear systems only
  • Not a CAS — no symbolic factoring or simplification
  • No transcendental equations or inequalities
  • Double-precision arithmetic can round on extreme inputs
  • Singular systems (zero determinant) flagged, not forced
Privacy & Security

Privacy and Offline Use

All solving runs in the browser as pure functions, so nothing you enter is sent to a server, there is no account or tracking, and the page keeps working offline after first load. Coefficients live only in the page and reset on reload.

Equation solving: in-browser (AnyTool) vs typical online solvers
CapabilityAnyToolTypical online solvers
Processing100% in-browser, offline-capableOften server-side
Equation typesLinear, quadratic, cubic, 2×2 & 3×3 systemsVaries
Step-by-step workingYes, for every modeOften paywalled
Complex rootsShown as a + biSometimes omitted
PrivacyNothing uploaded, no accountMay log inputs