[Math] Discriminant (in the context of PDE classification): $b^2 – 4ac$ or $b^2 – ac$

Question

I'm reading two textbooks on partial differential equations. In their respective sections on classification of PDEs (hyperbolic, parabolic, elliptical), they differ in what they describe as being the discriminant. One textbook says that the discriminant is $b^2 – 4ac$, while the other describes it being $b^2 – ac$. Are these both correct, or is one correct and the other incorrect?

EDIT: What are the $a$, $b$, and $c$? The second-order linear PDE in two independent variables is $Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G$, where $A$, $B$, $C$, $D$, $E$, $F$, $G$ are functions of $x$ and $y$ and could be constants.

Answer 1

Some authors/texts like to write the middle term as $2Bu_{xy}$ as opposed to $Bu_{xy}.$ In that case, it is $B^2-AC$ and in your case, it should be $B^2-4AC.$