Root Tool
Calculate indexed roots of real numbers or solve polynomial equations from degree two to six. Explore real and complex solutions, choose decimal precision, inspect a zoomable function graph, review numerical details, and export results.
Table of Contents
Highlights
- Indexed roots from 1st to 100th degree
- Real and complete complex-root options
- Polynomial roots for degrees 2 to 6
- Interactive zoomable graph with coordinate inspection
- CSV, JSON, TXT and SVG exports
How Root Tool Works?
An indexed root solves an equation of the form xn = a, where a is the entered number and n is a whole-number index from 1 to 100. For a positive number and even index, there are two real solutions, one positive and one negative. An odd index gives one real solution for any real number. A negative number with an even index has no real solution.
Choosing all complex roots finds solutions around a circle in the complex plane. For non-zero a, their angles are evenly separated; when a is zero, the tool displays zero once.
The polynomial mode accepts real coefficients for degrees 2 through 6, entered from the highest power to the constant term. It finds all roots, including complex roots, by Durand–Kerner iteration followed by Newton refinement. The leading coefficient must be non-zero.
| Input or option | Effect |
|---|---|
| Number and root index | Solve xn = a. |
| Real or all complex roots | Restrict to real solutions or include complex ones. |
| Polynomial degree and coefficients | Solve a degree 2–6 polynomial, including complex solutions. |
| Decimal places | Display 2, 4, 6, 8, 10 or 12 decimal places. |
| Graph view span | Zoom the real-valued function plot; hover or tap to inspect coordinates. |
Results include a root table, magnitudes for indexed roots or residuals for polynomial roots, and an interactive graph of y = xn − a or y = f(x). Results can be exported as CSV, JSON, TXT or graph SVG.
Practical Use Cases
Students: Compare square, cube and higher-order roots, including why negative inputs behave differently with odd and even indices. Use the function graph to identify real x-intercepts.
Algebra teachers: Demonstrate the difference between real and complex solutions and explore how polynomial coefficients change root locations.
Engineering students: Analyse quadratic through sextic characteristic equations, review complex root pairs and inspect the numerical residual at each computed solution.
Researchers and technical writers: Check small polynomial examples, control displayed decimal places and export root lists as CSV, JSON or TXT for documentation.
Visual learners: Adjust the graph's viewing span and hover or tap the curve to see how the function changes near a root.
Accuracy and Limitations
Calculations use JavaScript floating-point arithmetic, so displayed digits beyond numerical precision are not guaranteed. Indexed roots support integer indices from 1 to 100; even roots of negative numbers require the complex option. Polynomial solving supports real coefficients and degrees 2–6 only, with a non-zero leading coefficient and coefficient magnitudes no greater than 1080. The iterative solver may fail for ill-conditioned or tightly clustered roots; it rejects solutions with excessive residuals. Repeated roots can be numerically sensitive. The graph shows real x-values only, so non-real roots are listed but not plotted as points. Decimal-place settings change presentation rather than internal precision.
