Just Touching

Sometimes the quadratic you get has only one solution. Take y = x^2 and y = 2x - 1. Substituting gives x^2 = 2x - 1, so x^2 - 2x + 1 = 0, which factors as (x - 1)^2 = 0. The only x is 1, and the line gives y = 1: one meeting point, (1, 1).

A line like this is a tangent: it touches the parabola at one point without crossing it. Everywhere else the parabola stays above the line, because the difference between them is the square of x - 1, which is never negative.

The discriminant sees this coming. x^2 - 2x + 1 = 0 has discriminant 4 - 4, which is 0: one repeated solution, so one meeting point. A positive discriminant would have meant two crossings, as in the last lesson.