the discriminant of a PDE
You may remember from school that a quadratic ax^2 + bx + c is sorted by the number b^2 - 4ac: positive means two real roots, zero means a repeated root, negative means none. The discriminant of a second-order PDE is the same idea wearing different clothes — a single number, computed from the leading coefficients, that tells you in advance which of three deeply different worlds the equation lives in. It is the one quantity you compute first when you meet a second-order linear PDE.
For the equation A u_xx + 2B u_xy + C u_yy + (lower order) = 0, the discriminant is B^2 - AC, built only from the principal-part coefficients. (Watch the convention: because the cross term is written 2B u_xy, the natural quantity is B^2 - AC, not B^2 - 4AC. If someone writes the cross term as just B u_xy, their discriminant becomes B^2 - 4AC. Same idea, different bookkeeping.) The sign of B^2 - AC is the whole point: B^2 - AC < 0 is elliptic, B^2 - AC = 0 is parabolic, B^2 - AC > 0 is hyperbolic. Geometrically, set up the quadratic form A p^2 + 2B p q + C q^2; its level set is an ellipse, parabola, or hyperbola exactly as B^2 - AC is negative, zero, or positive — which is where the three type names come from.
Why it deserves so much respect: the sign of one number predicts what boundary or initial data is well-posed, whether disturbances travel at finite or infinite speed, how smooth the solutions are, and whether real characteristic curves exist. Two honest cautions, though. First, when the coefficients depend on x and y, the discriminant can change sign from region to region, so the same equation can be elliptic in one place and hyperbolic in another — that is an equation of mixed type, like Tricomi's. Second, in more than two independent variables a single number no longer suffices; you classify by the signs of the eigenvalues of the coefficient matrix instead.
Wave: u_tt - u_xx = 0 gives A=-1, B=0, C=1, so B^2 - AC = 0 - (-1)(1) = 1 > 0, hyperbolic. Laplace: u_xx + u_yy = 0 gives A=1, B=0, C=1, so B^2 - AC = -1 < 0, elliptic. Heat: u_t - u_xx = 0 has only u_xx among second derivatives, so A=1 (on u_xx), B=C=0 and B^2 - AC = 0, parabolic.
One short computation per equation lands each of the three models in its type — the discriminant is genuinely the first thing to evaluate.
Mind the convention: with the cross term written 2B u_xy the discriminant is B^2 - AC; with it written B u_xy it is B^2 - 4AC. The classification (sign of the discriminant) is identical either way — only the formula's middle coefficient differs.