Quartic Equation Calculator

This quartic equation calculator solves any fourth degree polynomial equation of the form ax to the fourth plus bx cubed plus cx squared plus dx plus e equals zero, returning all four roots at once, whether they are real or complex. You enter the five coefficients, a, b, c, d and e, with a unable to be zero because that would turn the equation into a cubic rather than a quartic. Rather than relying on the notoriously long and error-prone general quartic formula, the calculator solves the equation numerically, using the Durand-Kerner method, to find every root to high precision. Results update as you type, showing a single list of roots, the number of real roots found, and the leading coefficient a you entered, so you can quickly see whether the equation has zero, two or four real solutions, with any remaining roots forming complex conjugate pairs. This makes it handy for checking algebra homework, verifying a factorisation, or solving equations that would be slow and fiddly to factor by hand. Simply enter your five coefficients, matching the powers of x from highest to lowest, and read the roots off straight away. Because the method is numerical, values are rounded for display, so treat any real or imaginary part that is extremely close to zero as effectively zero when comparing against your own working.

Roots: 3, -3, -1, 1

Real roots4

Leading coefficient1

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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