Quadratic polynomial

A quadratic polynomial has total degree exactly two, so at least one degree-two term has a nonzero coefficient and no term has degree greater than two. ¹ In one variable, the standard form is f(x) = ax² + bx + c, with a ≠ 0. The coefficients may belong to the real numbers, complex numbers, or another specified field.

With variables x₁, …, x_n, a quadratic polynomial may include squared terms and pairwise products such as x₁² and x₁x₂, together with linear and constant terms. In matrix notation it can be written x^TAx + b^Tx + c, where the quadratic part is a quadratic form. The zero polynomial is not quadratic because its degree is undefined or conventionally assigned a value below two.

A univariate quadratic polynomial is governed by its leading coefficient, discriminant, and factorization. The associated quadratic equation ax² + bx + c = 0 has solutions given by the quadratic formula, x = (−b ± √(b² − 4ac))/(2a). ² The discriminant Δ = b² − 4ac determines the roots over the real numbers: positive Δ gives two distinct real roots, zero gives one repeated real root, and negative Δ gives a complex-conjugate pair.

As a function on the real line, the graph is a parabola. Its axis of symmetry is x = −b/(2a), and its vertex is the point where the polynomial attains a minimum when a > 0 or a maximum when a < 0. Completing the square exposes this geometry and is also useful for integration, optimization, and numerical analysis.

In several variables, the quadratic part determines the principal geometry, while linear and constant terms translate it. A real quadratic polynomial in two variables can define a conic section through an equation such as Ax² + Bxy + Cy² + Dx + Ey + F = 0. Depending on its coefficients, the resulting locus can be an ellipse, parabola, hyperbola, a degenerate pair of lines, a point, or have no real points. ³

For f(x) = x^TAx + b^Tx + c, only the symmetric part of A affects the polynomial, since x^TAx = x^T((A + A^T)/2)x over fields whose characteristic is not two. Orthogonal changes of coordinates can diagonalize a real symmetric matrix, reducing the quadratic part to a sum of scaled squares. This connects quadratic polynomials with eigenvalues, curvature, and quadratic optimization.

Quadratic polynomials also appear as local approximations and as algebraic objects whose behavior depends strongly on the coefficient domain. In multivariable calculus, the second-degree Taylor polynomial uses the Hessian matrix to capture local curvature; its quadratic part helps classify stationary points as locally minimizing, maximizing, or saddle-like. ⁴

Over the real numbers, a quadratic form is classified partly by the signs of its eigenvalues: positive-definite forms are positive away from the origin, while indefinite forms take both positive and negative values. This distinction matters in least-squares methods, mechanics, statistics, and convex optimization. ⁵ Over a field of characteristic two, division by two is unavailable, so symmetric-matrix descriptions and familiar completing-the-square arguments require modification. A quadratic polynomial is therefore not merely a formula with a squared variable; its algebraic properties depend on variables, coefficients, transformations, and the underlying number system.

Degree

The greatest total exponent of any nonzero monomial in a polynomial.

Discriminant

For ax² + bx + c, the quantity b² − 4ac, which describes the nature and multiplicity of its roots.

Quadratic form

A homogeneous polynomial of degree two, commonly written xᵀAx.

Hessian matrix

The matrix of second partial derivatives of a multivariable function.

Conic section

A curve obtained as the intersection of a plane with a cone, including ellipses, parabolas, and hyperbolas.