A quadratic form is a type of mathematical expression that involves variables raised to the second power, typically represented as ax^2 + bxy + cy^2 . Here, a , b , and c are constants, while x and y are variables. Quadratic forms can be used to describe various geometric shapes, such as conic sections, which include circles, ellipses, parabolas, and hyperbolas. Quadratic forms can be analyzed using matrices, where they are often expressed in the form \mathbfx^T \mathbfA \mathbfx , with \mathbfx being a vector of variables and \mathbfA a symmetric matrix. This representation helps in studying properties like definiteness and eigenvalues, which are important in fields such as optimization and statistics.