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 ${\left(x-1\right)}^{2} = 0$. 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.