classification by the symbol
The discriminant B^2 - AC is a wonderful sorting hat, but it only works for two variables. The real world routinely has three space dimensions plus time, so you need a classifier that does not care how many variables there are. That classifier is the principal symbol read as a quadratic form: count the signs of its eigenvalues, and you have named the type in any dimension. This is the grown-up version of the trichotomy.
For a second-order operator in n variables, the principal symbol is a quadratic form xi-transpose times A times xi, where A is the symmetric matrix of the leading coefficients (A_{ij} sits on the u_{x_i x_j} term). Diagonalize A and look at the signs of its eigenvalues. Elliptic: all eigenvalues the same sign (the form is definite) — the Laplacian u_xx + u_yy + u_zz, all positive. Parabolic: one eigenvalue is zero and the rest share a sign (the form is degenerate but semi-definite), the situation of the heat operator where the time direction contributes no second derivative. Hyperbolic: all eigenvalues non-zero, with exactly one differing in sign from the rest (one minus, the others plus), as in the wave operator u_tt - u_xx - u_yy - u_zz. And the case the two-variable picture cannot show: ultrahyperbolic, with two or more eigenvalues of each sign and none zero.
This is why the symbol is the honest, general definition of type — the two-variable discriminant is just the special case n = 2, where a 2-by-2 symmetric matrix has eigenvalue signs encoded in det(A) = AC - B^2, whose sign is the opposite of the discriminant's. Once you classify by the symbol, everything generalizes: elliptic operators in any dimension are smoothing and want boundary data, hyperbolic ones propagate at finite speed and want Cauchy data, parabolic ones diffuse. Two honest caveats: with variable coefficients the eigenvalue signs can change from point to point (mixed type), and this clean eigenvalue test is the classification for SECOND-order operators — higher-order operators need the more general definition that elliptic means the principal symbol vanishes only at xi = 0.
For u_tt - u_xx - u_yy - u_zz = 0 (the 3+1 wave equation), the coefficient matrix of the principal symbol is diagonal with entries (1, -1, -1, -1): one positive, three negative — exactly one odd sign — so the equation is hyperbolic in four variables, just as the discriminant would have said in two.
Beyond two variables, count eigenvalue signs of the principal-symbol matrix: all same = elliptic, one odd = hyperbolic, one zero = parabolic, several of each = ultrahyperbolic.
The two-variable discriminant B^2 - AC is just this eigenvalue test in disguise (for a 2-by-2 matrix the eigenvalue signs are encoded in AC - B^2); the symbol is the general, dimension-free statement and is the only correct way to classify in three or more variables.