Discriminant and x-axis intersections

The number of points a parabola shares with the $x$-axis is decided by the sign of the discriminant $D = b^2 - 4ac$: two points when $D > 0$, one point (tangent) when $D = 0$, and none when $D < 0$.

The three parabolas on the graph compare the cases. $y = x^2 - 2x - 3$ has $D = 16 > 0$ and crosses at $x = -1, 3$. $y = x^2 - 2x + 1 = (x - 1)^2$ has $D = 0$ and touches the $x$-axis at $(1, 0)$. $y = x^2 - 2x + 2$ has $D = -4 < 0$ and never meets the $x$-axis. The large dots mark the shared points.