Quartic Equation Calculator
This quartic equation calculator solves ax⁴ + bx³ + cx² + dx + e = 0 for any real coefficients: it lists all four roots – real and complex, with multiplicity – and shows the discriminant Δ with the case it stands for. Biquadratic equations are solved directly via y = x².
Example: a (coefficient of x⁴, not 0) 1 · b (coefficient of x³) -10 · c (coefficient of x²) 35 · d (coefficient of x) -50 · e (constant term) 24 → Real roots: Real roots: x = 1; 2; 3; 4. Source: Wikipedia, Quartic function – Nature of the roots: discriminant Δ, P = 8ac − 3b², D = 64a³e − 16a²c² + 16ab²c − 16a²bd − 3b⁴; Δ < 0 two real and two complex roots, Δ > 0 with P < 0 and D < 0 four distinct real roots, otherwise two complex pairs; biquadratic case (read 15.10.2026). Updated: .
How it is calculated
A quartic equation has exactly four roots when complex roots and multiplicities are counted. x⁴ − 10x³ + 35x² − 50x + 24 = 0 factors as (x − 1)(x − 2)(x − 3)(x − 4), so its roots are 1, 2, 3 and 4; its discriminant is Δ = 144.
Examples
| a | b | c | d | e | Real roots |
|---|---|---|---|---|---|
| 1 | −10 | 35 | −50 | 24 | 1; 2; 3; 4 |
| 1 | 0 | −5 | 0 | 4 | −2; −1; 1; 2 |
| 1 | 0 | 1 | 0 | −6 | −1.4142; 1.4142 |
| 1 | 0 | 0 | 0 | 1 | none |
The third equation also has the complex pair ±1.7321i (x² = −3); the last one has four complex roots ±0.7071 ± 0.7071i.
What the discriminant says
- Δ < 0: two real roots and two complex ones.
- Δ > 0 with P < 0 and D < 0: four different real roots.
- Δ > 0 with P > 0 or D > 0: no real root, two complex pairs.
- Δ = 0: a root occurs twice or more, as in (x − 1)⁴ = x⁴ − 4x³ + 6x² − 4x + 1.
Biquadratic equations
Without x³ and x terms (b = d = 0) substitute y = x²: x⁴ − 5x² + 4 = 0 becomes y² − 5y + 4 = 0 with y = 1 and y = 4, so x = ±1 and x = ±2. The quadratic step works like the quadratic formula calculator; one degree lower there is the cubic equation solver.
Frequently asked questions
How do you solve a quartic equation?
Ferrari’s method reduces it to a cubic and two quadratics. In practice roots are found numerically, as here. For x⁴ − 10x³ + 35x² − 50x + 24 = 0 the roots are 1, 2, 3 and 4.
How many roots does a quartic equation have?
Exactly four, counting complex roots and multiplicity. It can have 4, 2 or 0 real roots.
What is a biquadratic equation?
A quartic without x³ and x terms: ax⁴ + cx² + e = 0. With y = x² it becomes quadratic.
What does the discriminant of a quartic tell you?
Its sign gives the nature of the roots: Δ < 0 means two real and two complex roots, Δ = 0 a multiple root, Δ > 0 either four real or no real roots.
Can the calculator handle decimal coefficients?
Yes, any real numbers with a ≠ 0. The roots are shown to 8 significant digits.
Sources and legal basis
As of: · Editorial policy
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