the general second-degree equation
Every conic — circle, ellipse, parabola, hyperbola — and every tilted or shifted version of one can be written as a single algebraic template. That template is the general second-degree equation in x and y, and recognizing it lets you tell at a glance that an equation describes a conic, even when it looks like an unrecognizable mess.
It is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, where A, B, C, D, E, F are constants and at least one of A, B, C is nonzero (so there really is a squared term). Each piece has a job: A and C control the x^2 and y^2 stretching; the cross term Bxy tilts the conic so its axes no longer line up with the coordinate axes; D and E shift the center away from the origin; F sets the overall scale. If B = 0, the conic's axes are parallel to the coordinate axes and you can read it off by completing the square. If B is nonzero, the conic is rotated, and you remove the Bxy term by rotating the axes through a suitable angle. The discriminant B^2 - 4AC then tells you which conic you have without doing any of that work.
This one equation is the meeting point of algebra and geometry that Descartes envisioned: a purely symbolic object whose six coefficients encode a curve's type, size, position, and tilt. It also quietly includes the degenerate cases — a point, a single line, a pair of lines, or no real points at all — which arise for special coefficient values, so 'second-degree equation' is slightly more general than 'conic'.
The circle x^2 + y^2 = 25 is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 with A = C = 1, B = 0, D = E = 0, F = -25. The tilted curve xy = 1 has A = C = 0, B = 1, D = E = 0, F = -1 — only the cross term, so it is a rotated (rectangular) hyperbola. And 9x^2 + 4y^2 - 18x + 16y - 11 = 0 completes the square to an ellipse centered at (1, -2).
Six coefficients A..F encode a conic's type, size, position, and tilt all at once.
Not every second-degree equation is a (real, non-degenerate) conic: x^2 + y^2 + 1 = 0 has no real solutions, and x^2 - y^2 = 0 is the pair of lines y = +-x. The form is necessary but not sufficient; the discriminant and a degeneracy check finish the job.