# 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.*

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## 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

- 1933: D.M.Y. Sommerville:
*Analytical Conics*(3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $14$. Condition that the general quadratic equation of the second degree should represent two straight lines