Quartic Equation Calculator
This quartic equation calculator solves a fourth degree equation, ax to the fourth plus bx cubed plus cx squared plus dx plus e equals zero. Enter the five coefficients and the calculator returns all four roots, real or complex. Quartics can have zero, two or four real roots, with any remaining roots forming complex conjugate pairs. The general quartic formula is notoriously unwieldy, so the calculator solves the equation numerically to high precision, which is reliable and handles every case the same way. The method, a worked example and the assumptions are below.
The formula
A quartic has four roots, counted with multiplicity. The calculator finds them numerically with the Durand-Kerner method, returning real and complex roots together. Real roots come in any even or odd count up to four, with complex roots in conjugate pairs.
Worked example
For x^4 minus 10x^2 plus 9 = 0, treat it as a quadratic in x squared: (x^2 minus 1)(x^2 minus 9) = 0, giving roots 1, minus 1, 3 and minus 3. Enter 1, 0, -10, 0, 9 to confirm.
Frequently asked questions
How many roots does a quartic have?
Four, counted with multiplicity, with complex roots appearing in conjugate pairs.
Why solve numerically?
The exact quartic formula is extremely long and error prone. A numerical solver is accurate and treats every case uniformly.
Can a quartic have no real roots?
Yes. If all four roots are complex, the curve never crosses the x axis.
Who this calculator is for
This calculator is for algebra students and anyone needing a quick, reliable result.
What this calculator assumes
- You enter valid numbers.
- The standard method is applied.
- Results are rounded for display.
Formula and sources
Related calculators
- Quadratic Formula Calculator: solve a quadratic.
- Scientific Calculator: general maths.
- Matrix Calculator: matrix operations.
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