Build the idea in three steps
- For a quadratic equation ax² + bx + c = 0 with real coefficients and a ≠ 0, the sign of b² − 4ac determines the number of real roots: two, one or none. A zero discriminant means a repeated root.
- A cubic curve in the plane can fail the same way: at a singularity, the curve crosses itself or comes to a point. That is not a drawing blemish. The arithmetic arguments used here break down exactly at such points.
- The Weierstrass discriminant Δ gathers auxiliary quantities b₂, b₄, b₆ and b₈ computed from the five coefficients. When the coefficient type R is a field, Δ = 0 holds exactly when the curve has a singularity.
A small example
For y² = x³ − x, Δ = 64, so the curve is nonsingular over ℚ. For y² = x³, Δ = 0 and (0,0) is singular. This concerns the curve equation; it does not say that every pair of points merges.