Solve fourth-degree polynomial equations of the form ax⁴ + bx³ + cx² + dx + e = 0.
Input Parameters
Roots (Solutions)
2.0000
Root 1
Real and complex roots found using numeric methods.
Root 2-2.0000
Root 31.0000
Root 4-1.0000
About this calculator
Quick Reference
Degree 4Quartic
Max Real Roots4
SolvableYes (Ferrari's Method)
What is a Quartic Equation?
A quartic equation is an algebraic equation of the fourth degree. It is the highest degree polynomial equation that can be solved by radicals in the general case (Abel-Ruffini theorem states that equations of degree 5 and higher cannot). Lodovico Ferrari is credited with discovering the solution to the quartic in 1540.
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Pro Tips
Degree: There are exactly 4 roots (counting multiplicity and complex roots).
Symmetry: If all coefficients are real, complex roots always appear in conjugate pairs (a+bi and a-bi).
Complexity: The general algebraic solution is extremely lengthy, so numerical methods are often preferred.
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Fun Facts
"Lodovico Ferrari was a student of Gerolamo Cardano, who published the solution in his book 'Ars Magna'."
"Quartic equations are used in computer graphics for ray-tracing intersections with tori and other shapes."
