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or, equivalently,

Remark There are other methods for eliminating the cross-product terms from a quadratic form; we shall not discuss them
here. Two such methods, Lagrange's reduction and Kronecker's reduction, are discussed in more advanced books.

Conic Sections

We shall now apply our work on quadratic forms to the study of equations of the form
(2)

where a, b, …, f are real numbers, and at least one of the numbers a, b, c is not zero. An equation of this type is called a
quadratic equation in x and y, and
is called the associated quadratic form.

EXAMPLE 2 Coefficients in a Quadratic Equation
In the quadratic equation
the constants in 2 are

EXAMPLE 3 Examples of Associated Quadratic Forms

Quadratic Equation Associated Quadratic Form

Graphs of quadratic equations in x and y are called conics or conic sections. The most important conics are ellipses, circles,
hyperbolas, and parabolas; these are called the nondegenerate conics. The remaining conics are called degenerate and include