Definition:Discriminant of Quadratic Equation in Two Variables

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This page is about Discriminant of Quadratic Equation in Two Variables. For other uses, see Discriminant.

Definition

Consider the quadratic equation in $2$ variables:

$(1): \quad a x^2 + b y^2 + 2 h x y + 2 g x + 2 f y + c = 0$

where $x$ and $y$ are independent variables.


The discriminant of $(1)$ is the expression:

$a b c + 2 f g h - a f^2 - b g^2 - c h^2$


Also presented as

It can also be expressed in the form of a determinant:

$\begin {vmatrix} a & h & g \\ h & b & f \\ g & f & c \end {vmatrix}$


Sources