Polynomial Roots Calculator
This calculator finds the roots (zeros) of quadratic and cubic polynomials using exact analytic formulas. For quadratics ax^2 + bx + c = 0, it applies the quadratic formula. For cubics ax^3 + bx^2 + cx + d = 0, it uses Cardano's method: substituting x = t - b/(3a) to remove the quadratic term, reducing to the depressed cubic t^3 + pt + q = 0, then applying the cube root formula with the discriminant. All real and complex roots are shown. Select the polynomial degree, enter the coefficients, and click to compute.
Polynomial root formulas
Quadratic: x = (-b +/- sqrt(b^2 - 4ac)) / (2a)
Cubic: reduce to t^3 + pt + q = 0 via x = t - b/(3a)
D = -(4p^3 + 27q^2); Cardano gives cube root solutions
For quadratics the discriminant D = b^2 - 4ac determines the root type. For cubics, after removing the quadratic term, the discriminant of the depressed cubic determines whether all roots are real or one is real and two are complex.
Interpreting polynomial roots
- Each root r satisfies the polynomial equation: substituting r for x yields zero.
- Real roots correspond to x-intercepts where the polynomial graph crosses the x-axis.
- Complex roots always appear in conjugate pairs for polynomials with real coefficients.
- A repeated root (multiplicity 2) means the graph touches but does not cross the x-axis.
- Vieta's formulas: for a quadratic, sum of roots = -b/a and product of roots = c/a.
Polynomial roots: frequently asked questions
How are quadratic roots found?
Quadratic roots ax^2 + bx + c = 0 are found with x = (-b +/- sqrt(b^2-4ac))/(2a). The discriminant b^2-4ac determines the nature: positive gives two real roots, zero gives one repeated root, negative gives two complex conjugate roots.
How are cubic roots found?
A depressed cubic t^3 + pt + q = 0 is solved with Cardano's formula. The general cubic ax^3+bx^2+cx+d is first reduced to a depressed form by the substitution x = t - b/(3a). The discriminant of the cubic determines whether there are three real roots or one real and two complex roots.
What is the discriminant of a cubic?
For x^3 + px + q, the discriminant is D = -(4p^3 + 27q^2). If D > 0, there are three distinct real roots (casus irreducibilis). If D = 0, there is at least one repeated root. If D < 0, there is one real root and two complex conjugate roots.
Can every polynomial be solved by a formula?
Degree 1 and 2 have simple formulas. Degree 3 and 4 have general formulas (Cardano and Ferrari). Abel's impossibility theorem proves that no general formula using radicals exists for degree 5 or higher. Numerical methods are used instead.
What does it mean for a polynomial to have complex roots?
Complex roots come in conjugate pairs (a + bi and a - bi) for polynomials with real coefficients. They correspond to quadratic factors with negative discriminant. The polynomial does not cross the real x-axis at those roots.
Official sources
- NIST Digital Library of Mathematical Functions, Chapter 1 (Algebraic and Analytic Methods): dlmf.nist.gov/1.
- NIST, Mathematical topics: nist.gov/topics/mathematics.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.