[Math] formula for the checking of the PDE to be hyperbolic, elliptic, parabolic

Question

$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$
the formula that my book uses to check for the conditions for the elliptic, hyperbolic and the parabolic equation is
$$b^2-4ac\gt 0$$
for hyperbolic equations,
$$b^2-4ac\lt 0$$
for elliptic equations and
$$b^2-4ac=0$$
for the parabolic equation.
But on wiki it is mentioned to be by using
$$b^2-ac,$$
is the result given in my book is wrong.
What is the soul derivation of saying about these equations to be hyperbolic, parabolic, elliptic?

Answer 1

The definition in Wikipedia is correct and it is reffer to an equation written in the form: $$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$$ , but I suspect that also the definition in you book is correct and it is referred to an equation written in the form: $$Au_{xx} + B'u_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$$

Note that, in this case, we have: $$ (B')^2-4AC=(2B)^2-4AC=4(B^2-AC) $$