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Math calculator

Numerical Root Calculator

Find a real root of a polynomial with Newton, bisection, or secant. One capability, three methods. Not the nth-root page. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Specification checked · 9/9 tests · Production surface contract 1/1 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
A real polynomial root by Newton, bisection, or secant.
Scope
Real polynomials
Verification
Engine tested · 9/9 tests · Production surface contract 1/1 · Specification checked · v1.0.0
Named expert review
Optional · Not performed
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Specification basis
Evidence
2 golden · 2 boundary · 5 property · Production surface contract 1/1 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status PASS (0 stale; 164 CURRENT) @ 2026-09-16T06:44:24.909Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Newtonx ← x − f(x)/f′(x)
Bisectionmidpoint of a sign-change bracket
Secantx ← x₁ − f(x₁)(x₁−x₀)/(f(x₁)−f(x₀))
iCoefficients are constant-first (c0,c1,…). Degree ≤ 8. This is not ⁿ√a — that remains /calc/math/root.

How to use

1

Enter coefficients constant-first

x² − 2 is −2,0,1. Newton needs x0; bisection needs a and b with opposite signs.

2

Read the approximate root

Residual and iteration count are in the result.

Example calculations

Common configurations with formula and result.

ϟ

√2

x² − 2 = 0, Newton from 1.5

x ← x − (x²−2)/(2x)
1.414213…
ϟ

Linear

x − 4 = 0

coeffs −4,1
4

Numerical Root calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A real root of f(x) = c₀ + c₁x + … + cₙxⁿ is a number r with f(r) = 0. Newton, bisection, and secant are methods of this capability — not three product pages.
What it calculates
A real polynomial root by Newton, bisection, or secant.
Inputs
  • method
  • coeffs
  • x0?
  • x1?
  • a?
  • b?
Outputs
  • root
  • residual
  • iterations
Formula
f(x) = Σ cₖ xᵏ
Assumptions
  • Real polynomials
  • Not CAS / not complex roots listed
Units
  • dimensionless
Boundary conditions
  • no sign change → INVALID_INPUT
  • unknown method → INVALID_MODE
  • no convergence → NO_CONVERGENCE
Example
newton coeffs=-2,0,1 x0=1.5 → √2
Validation cases

2 published on this page · 9/9 tests · Production surface contract 1/1 · View evidence

  • method=newton coeffs=-2,0,1 x0=1.5 → root≈1.414213562
  • method=halley coeffs=-2,0,1 x0=1 → INVALID_MODE
Specification basis
  • ISO 80000-2:2019 Quantities and units — Mathematics
Calculation version
1.0.0

Other calculators in this family: Root (ⁿ√a), Quadratic, Definite integral .

ⁿ√ Root (ⁿ√a) › Δ Quadratic › ∫ Definite integral ›
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Frequently asked questions

Key distinctions behind the calculation.

Is this the nth-root calculator?

No. ⁿ√a is /calc/math/root. This page finds zeros of a polynomial.

Why not three pages for Newton, bisection, and secant?

They are methods of one model. Switching method does not change the capability.