Quartic Polynomial Input Form
Use the responsive calculator grid below. Large screens show three columns, medium screens show two, and mobile shows one.
Example Data Table
This example uses the quartic polynomial x4 - 5x2 + 4 = 0.
| a | b | c | d | e | Roots | Critical x-values |
|---|---|---|---|---|---|---|
| 1 | 0 | -5 | 0 | 4 | -2, -1, 1, 2 | -1.581139, 0, 1.581139 |
Formula Used
The calculator solves a general quartic equation of the form ax⁴ + bx³ + cx² + dx + e = 0.
Using x = y - b/(4a), the equation becomes y⁴ + py² + qy + r = 0, where p, q, and r are displayed in the results.
Roots are computed with the Durand–Kerner method, which iteratively refines all polynomial roots together and handles real and complex solutions.
Critical points come from the derivative: f′(x) = 4ax³ + 3bx² + 2cx + d. The calculator solves this cubic numerically.
Inflection points come from the second derivative: f″(x) = 12ax² + 6bx + 2c. Real solutions are shown in the results table.
For ax⁴ + bx³ + cx² + dx + e = 0: sum of roots = -b/a, pairwise sum = c/a, triple sum = -d/a, product = e/a.
How to Use This Calculator
- Enter the five coefficients a, b, c, d, and e.
- Keep a nonzero, because a quartic needs an x⁴ term.
- Optionally enter a value for x to evaluate the function.
- Click the calculate button to generate roots and graph data.
- Review real roots, complex roots, turning points, and inflection points.
- Use the CSV button for spreadsheet export of result metrics.
- Use the PDF button to save a clean report section.
- Try the example loader for a known polynomial with four real roots.
Frequently Asked Questions
1. What does this quartic calculator solve?
It solves equations of the form ax⁴ + bx³ + cx² + dx + e = 0, then reports roots, critical points, inflection points, Vieta values, and a graph.
2. Can it show complex roots?
Yes. When a quartic has nonreal solutions, the calculator lists them in a + bi format. Real roots and complex roots are also counted separately.
3. Why must coefficient a be nonzero?
If a equals zero, the equation is no longer quartic. It becomes a lower-degree polynomial, which changes the solving method and graph behavior.
4. How are the roots computed?
The code uses the Durand–Kerner numerical method. It refines all roots at once and is effective for quartic equations with mixed real and complex solutions.
5. What are critical points in this report?
Critical points are x-values where the first derivative equals zero. They often represent local maxima, local minima, or stationary inflection behavior.
6. What are inflection points?
Inflection points are where the curve changes concavity. The calculator finds them by solving the second derivative and then evaluating the original polynomial.
7. What does the PDF export include?
The PDF captures the visible result report, including key values, tables, and the graph area. It is useful for saving or sharing a clean calculation summary.
8. Can I use this for coursework or checking answers?
Yes. It works well for learning, verification, graph inspection, and quick analysis. You should still show manual steps if your assignment requires full derivations.