What the calculator does
Enter the five coefficients of a quartic equation and the solver returns:
- all four roots, counted with multiplicity, whether real or complex;
- exact values for rational roots and for roots of quadratic factors;
- the discriminant and an exact count of the real roots;
- the factored form over the rational numbers;
- step-by-step working.
How to use it
Write the equation as ax⁴ + bx³ + cx² + dx + e = 0 and enter the five coefficients. Use 0 for any missing term: for x⁴ − 16 = 0, enter a = 1, b = 0, c = 0, d = 0, e = −16. Decimals and fractions are treated as exact values.
The method
1. Look for rational roots
By the rational root theorem, a rational root p/q of an integer-coefficient quartic has p dividing e and q dividing a. Each root found gives a linear factor, and dividing it out leaves a cubic or quadratic.
2. Look for quadratic factors
A quartic without rational roots may still split into two quadratics with rational coefficients. The solver finds such factors by grouping the roots in pairs and confirms each one by exact polynomial division.
3. Ferrari’s method
Otherwise, dividing by a and substituting x = y − b/(4a) gives the depressed quartic
If q = 0, this is a quadratic in y². If not, Ferrari’s idea is to add a parameter m so that both sides become perfect squares. That works when m is a root of the resolvent cubic
With s = √(2m), the quartic then factors into two quadratics:
The discriminant
| Value of Δ | Roots |
|---|---|
| Δ > 0 | four distinct roots, either all real or all non-real |
| Δ = 0 | at least one repeated root |
| Δ < 0 | two distinct real roots and a pair of complex conjugates |
Worked example
Solve x⁴ − 5x² + 4 = 0.
- Spot the structure. There are no odd powers, so substitute z = x² to get z² − 5z + 4 = 0.
- Solve the quadratic. (z − 1)(z − 4) = 0, so z = 1 or z = 4.
- Take square roots. x² = 1 gives x = ±1; x² = 4 gives x = ±2.
The roots are -2, -1, 1, 2, and the discriminant is 5184, which is positive, consistent with four distinct real roots.
Common mistakes
- Taking only the positive square root. In a biquadratic, each positive value of x² gives two roots, ±√z.
- Forgetting complex roots. A negative value of z = x² still gives two roots, ±i√|z|.
- Assuming Δ > 0 means four real roots. It can also mean no real roots at all.
- Dividing by x without recording x = 0. If e = 0, x = 0 is a root; record it before dividing.
Practical applications
- Geometry. Intersections of two conic sections, such as an ellipse and a circle, lead to quartic equations.
- Computer graphics. Ray tracing a torus requires solving a quartic for each ray.
- Mechanics. Some beam deflection and orbit problems lead to quartics.
- Linear algebra. The eigenvalues of a 4×4 matrix are roots of its characteristic quartic.
Assumptions and limitations
Coefficients must be real numbers. Rational roots and roots of quadratic factors are given exactly. Other roots are computed numerically to about 15 significant digits. The Ferrari working is computed independently and checked against these roots; if the two ever disagree, the solver shows the numerical result without the Ferrari steps rather than display working that does not match.