How to run each mode 01 Polynomial: type coefficients high to low, comma-separated — 1, -6, 11, -6 is x³ − 6x² + 11x − 6 — and press Solve. 02 System: set the size (2–6), fill the [A|b] grid or pick a matrix variable from the workspace, then Solve. 03 Roots: enter an expression in x, an interval a < b, and press Solve; the result lists x₁ ≈ …, each with |f(x)| and a warning line when a root is an even-multiplicity touch. 04 Read the verdicts. A polynomial degree 2 prints Δ and both roots; higher degrees come back as companion-matrix eigenvalues with a note that the method is approximate. 05 Send the answer on as a vector to the workspace, a report block, or another panel that accepts vectors. Worked readouts
Mode · input
Readout
Polynomial 1, -6, 11, -6
x1 = 1.00000000 , x2 = 2.00000000 , x3 = 3.00000000
Polynomial 1, 1e8, 1
x₁ = −100000000 , x₂ = −1e-8 , Δ = 9999999999999996
Polynomial 1, 0, 1
x₁ = 0.000000+1.000000i , x₂ = 0.000000-1.000000i
System 3×3, [A|b] rows 2 1 −1 8; −3 −1 2 −11; −2 1 2 −3
x1 = 2.00000000 , x2 = 3.00000000 , x3 = −1.00000000
Roots x^2 - 4, [−10, 10]
x1 ≈ −2 , x2 ≈ 2 , each |f(x)| = 0.00e+0
Roots sin(x), [−10, 10]
7 roots from x1 ≈ −9.424777961 to x7 ≈ 9.424777961
The cubic factors as (x−1)(x−2)(x−3), and the eigenvalues land on the integers to eight decimals. The 3×3 system satisfies all three rows: 2·2 + 3 − (−1) = 8, −6 − 3 + 2·(−1) = −11, −4 + 3 + 2·(−1) = −3.
Why the small root is right For x² + 10⁸·x + 1 the two roots differ by eight orders of magnitude, and the textbook formula loses most of the small root to cancellation: computing (−b + √Δ)/2a directly gives −7.45058e-9 , about 25.5 % away from the true −10⁻⁸. The panel takes the branch that avoids the subtraction — q = −(b + sign(b)·√Δ)/2 — and recovers the other root by Vieta's relation c/q, so it prints −1e-8 . The Δ row still shows the exact-looking 9999999999999996 , the double-precision value of 10¹⁶ − 4.
Limits
Degree ≥ 3 is numeric. Roots of higher-degree polynomials come from companion-matrix eigenvalues and are marked approximate; expect 1.00000000-style output, not symbolic forms. The leading coefficient must be non-zero and every coefficient must parse as a number.
Roots mode scans a finite interval. If no crossing exists in [a, b] it says so instead of guessing; sign changes at poles are excluded, and an even-multiplicity touch is labelled. What it doesn't do. It does not solve inequalities, systems with more than six unknowns, or symbolic equations with parameters, and it does not show a derivation — take the equation to step-by-step for factoring, completing the square and substitution checks, or to matrix tools for rank and determinants of the system matrix. The cancellation trap Type 1, 1e8, 1 and the panel answers x₂ = −1e-8 . The same quadratic solved with the naive formula on paper or in a spreadsheet gives −7.45e-9 — still close to zero, but a 25 % relative error on the root. The lesson generalises: when coefficients differ by many orders of magnitude, the displayed root is trustworthy only because the panel reorders the algebra; do not reproduce it by hand with the textbook expression.
Where it is useful Checking a fit or a design equation A calibration polynomial or characteristic equation can be pasted as coefficients and inspected root by root. The complex pair from 1, 0, 1 shows that no real root exists, so a real-valued design table has no solution to find.
Solving small equilibrium systems Three-node networks, balance equations and mesh currents reduce to 2×2–6×6 systems; the augmented grid accepts them directly, and a workspace matrix can be picked in one click. The LU solve prints eight decimals, which is enough to verify a hand calculation.
Privacy All three solvers run locally in your browser; coefficients, matrices and intervals are never uploaded.
References
Wikipedia, Quadratic equation , en.wikipedia.org （访问日期：2026-10-07）— discriminant and numerically stable root forms.
Wikipedia, Companion matrix , en.wikipedia.org （访问日期：2026-10-07）— roots as eigenvalues of the companion matrix.
Wikipedia, Brent's method , en.wikipedia.org （访问日期：2026-10-07）— bracketed root finding on an interval.
Wikipedia, System of linear equations , en.wikipedia.org （访问日期：2026-10-07）— augmented matrices and elimination.
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Reviewed by CalcX Editorial Team
Updated 2026-10-07