Find all real and complex roots of a polynomial up to degree 4 using exact algebraic methods.
This calculator uses exact algebraic methods for polynomials up to degree 4: the linear formula for degree 1, the quadratic formula for degree 2, Cardano's/trigonometric method for degree 3, and Ferrari's method for degree 4. All roots (real and complex) are reported.
Quadratic formula
x = (-b ± √(b²-4ac)) / (2a)
Cubic discriminant test
Δ = q²/4 + p³/27
There is no general closed-form solution for polynomials of degree 5 or higher (Abel-Ruffini theorem). Degrees 1-4 have exact algebraic formulas.
Complex roots appear as a±bi where i=√(-1). They always come in conjugate pairs for polynomials with real coefficients.