Equation Solver

About the Equation Solver

Type an equation in one unknown — 3(x − 2) + 4 = 2x + 7, x^2 − 5x + 6 = 0, (x + 1)/(x − 2) = 3, √(x + 3) = x − 3, 2^x = 10 or sin x = 0.5 — and get every solution, each one checked by putting it back into both sides. A graph of the two sides shows the solutions where the curves meet, and the steps say how they were found.

Polynomial and rational equations are solved exactly whenever possible: whole numbers and fractions, square roots such as (3 + √5)/2, cube roots, and complex roots such as (−1 + √3 i)/2 — with repeated roots and their multiplicity. Roots that make a denominator zero are shown and rejected as extraneous. When a polynomial has no simpler exact form, all its roots, real and complex, are found numerically. Other equations — with roots, absolute values, exponentials, logarithms or trigonometry — are solved on an interval you choose, by looking for sign changes and refining each one. Everything runs in your browser.

Common uses

• Checking homework answers for linear, quadratic and higher-degree equations — with the steps and the check.
• Finding where two functions are equal (where their graphs cross), such as e^x = 3x.
• Solving rational and radical equations and seeing which candidate roots are extraneous.
• Getting accurate numerical roots of polynomials of high degree, including the complex ones.

How exact solutions are found

Fractions are cleared by multiplying both sides by the lowest common denominator, which turns a rational equation into a polynomial P(x) = 0 (every root is then checked against the original denominators). The polynomial is split into repeated factors (square-free factorization), which gives each root’s multiplicity. Rational roots come from the rational root theorem: any root p/q has p dividing the constant term and q dividing the leading coefficient. A quadratic factor is solved with the quadratic formula, x = (−b ± √(b² − 4ac))/(2a), with the square root simplified; factors of degree 2–4 with whole-number coefficients are looked for too, and equations such as x⁴ − 10x² + 1 = 0 and x³ = 2 get nested or cube-root forms. What is left is solved with the Aberth–Ehrlich method (Aberth 1973), which finds all complex roots at once; Sturm’s theorem gives the exact number of real roots.

How other equations are solved

The equation is written as f(x) = left side − right side = 0, and f is evaluated at 20,001 evenly spaced points of the interval. Where f changes sign between two neighbouring points, Brent’s method — a safe mix of bisection, the secant method and inverse quadratic interpolation (Brent 1973, chapter 4) — narrows the root down to full precision. Places where |f| dips to 0 without changing sign (as for |x − 2| = 0 or sin x = 1) are found by minimising |f| — they count only if |f| really reaches 0 up to rounding, so cos x + 1.00000000001 = 0 correctly has no solution — and roots at the edge of the domain (such as √x = 0) are found where a side stops being defined. Where the two sides are equal on a whole stretch (|x| = x for every x ≥ 0), that stretch is given instead of single points. A sign change across a vertical asymptote, as tan x has at π/2, is recognised and rejected. Every root is then substituted back into both sides; decimals that match a simple value such as π/6, √2 or ln 3 are labelled.

How to type equations

• Powers: x^2, x², x^(1/3), 2^x, e^x (or exp(x)).
• Multiplication can be left out: 2x, 3(x + 1), (x + 1)(x − 1), x sin x.
• Roots: sqrt(x) or √x, cbrt(x) or ∛x, root(x, 5) for a fifth root.
• Absolute value: |x − 1| or abs(x − 1).
• Logarithms: ln x (natural), log x (base 10), log_2(x) or log2(x), any base with log_3(x).
• Trigonometry in radians: sin, cos, tan, sec, csc, cot, asin (or arcsin), sinh …; sin^2 x means (sin x)², sin^-1 x means arcsin x.
• Any single letter can be the unknown: 3t − 6 = 0, sin θ = 1.

Limitations

• One equation in one unknown, with real inputs. Systems of equations, inequalities and equations with complex coefficients are not supported.
• Non-polynomial equations are solved only on the chosen interval. Roots closer together than the sample spacing, or roots where both sides only nearly touch, can be missed; trigonometric equations usually have infinitely many solutions, of which only those in the interval are listed.
• Decimals reach from about 10⁻³⁰⁰ to 10³⁰⁸. Where every term is smaller than that (e^x for x below −745), the two sides cannot be told apart, so no solution is reported there and the page says so; where a side is larger, solutions cannot be searched for.
• Exact forms cover rational numbers, square roots (also nested ones for biquadratic equations), cube and fourth roots of x³ = c and x⁴ = c, and the complex roots that go with them. Other cubic and quartic equations are given as decimals rather than with the Cardano or Ferrari formulas. Polynomials can have degree up to 200; exact factoring is tried up to degree 60. When an expanded polynomial is very sensitive to rounding, as (x + 1)¹⁵⁰ = 2 is, roots that do not pass the check are left out and the page says how many.
• Decimals are shown to 10 significant digits (about 8 for a root where the graph only touches the axis). Roots in high-degree polynomials can be sensitive to their coefficients; each one shown has passed the check by substitution.

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