Solve quadratic, cubic and quartic polynomial equations instantly. Enter coefficients and get all real and complex roots with full step-by-step working shown — discriminant, factored form and verification.
Real & Complex RootsStep-by-Step
Polynomial Degree
x² − 5x + 6 = 0
Quadratic Formula Explained
For the equation ax² + bx + c = 0, the two roots are given by:
x = (−b ± √(b² − 4ac)) / (2a)
The expression under the square root, Δ = b² − 4ac, is called the discriminant and determines the nature of the roots.
The Discriminant and Nature of Roots
Discriminant Δ
Nature of Roots
Example
Δ > 0
2 distinct real roots
x² − 5x + 6 = 0 → x=2, x=3
Δ = 0
1 repeated real root
x² − 4x + 4 = 0 → x=2 (double)
Δ < 0
2 complex conjugate roots
x² + x + 1 = 0 → x=−½±(√3/2)i
Roots and Coefficients (Vieta's Formulas)
For any polynomial, the elementary symmetric functions of the roots equal the coefficients:
A root (or zero) of a polynomial p(x) is a value r such that p(r) = 0. Geometrically, real roots are where the curve y = p(x) crosses the x-axis. An nth-degree polynomial has exactly n roots in the complex number system (counted with multiplicity), according to the Fundamental Theorem of Algebra.
Cubic equations ax³+bx²+cx+d=0 can be solved by: (1) Rational Root Theorem — test ±(factors of d)/(factors of a) to find integer roots, then deflate to a quadratic. (2) Cardano's formula — a general algebraic formula for all three roots. (3) Newton-Raphson iteration — numerically converges to each root. This solver uses rational root testing first, then Newton-Raphson with synthetic division deflation.
Complex roots have the form a + bi where i = √(−1). For polynomials with real coefficients they always appear in conjugate pairs: if a+bi is a root, then a−bi is also a root. For a quadratic, complex roots arise when the discriminant b²−4ac is negative. They cannot be seen as x-axis crossings on a real graph, but they are genuine solutions in the complex number system.
Vieta's formulas relate a polynomial's coefficients to symmetric functions of its roots. For ax²+bx+c=0: sum of roots = −b/a and product = c/a. For a cubic ax³+bx²+cx+d=0: sum = −b/a, sum of pairwise products = c/a, product = −d/a. These allow quick verification of roots without full substitution.