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A quartic (fourth-degree) equation always has exactly four roots, but finding them by hand requires either a lengthy closed-form method (Ferrari's formula) or numerical approximation — there's no quick factoring shortcut for most quartics.
This solver finds all four roots of a quartic equation numerically, whether they're real or complex, real or repeated.
How it works
Enter the five coefficients (a through e) of \(ax^4+bx^3+cx^2+dx+e=0\). The solver normalizes the equation and applies the Durand-Kerner method, which iteratively refines all four roots simultaneously until they converge.
- Enter a (x⁴ coefficient).
- Enter b (x³ coefficient).
- Enter c (x² coefficient).
- Enter d (x coefficient).
- Enter e (constant).
- Click Calculate to see your results.
Examples
A quartic with four whole-number roots
The equation x⁴ - 10x³ + 35x² - 50x + 24 = 0 (which factors as (x-1)(x-2)(x-3)(x-4)) has roots 1, 2, 3, and 4.
Who should use it
- Solving a quartic equation from an algebra or precalculus course.
- Finding the roots of a quartic that arises from a physics, engineering, or optimization problem.
Industry applications
- Mathematics and engineering education
- Physics and optimization problems involving quartic models
Advantages
- Finds all four roots — real, complex, or repeated — with one consistent method.
- Handles the general quartic case without requiring the equation to be pre-factored.
Limitations
- Numerical convergence, not an exact symbolic formula — results are precise to several decimal places rather than exact fractions or radicals.
Common mistakes to avoid
- Entering a zero leading coefficient (a=0), which makes the equation cubic or lower, not quartic.
- Expecting all four roots to always be real — many quartics have two or four complex roots instead.
Best practices
- If you expect only real roots (e.g., from a known factored form), double-check the solver's output roots multiply back out to the original coefficients.
Tips
- If you already know one or two rational roots by inspection, you can factor those out first and solve the resulting lower-degree equation more simply — this solver is most useful when no roots are obvious.