Intersection of a quadratic function and the x-axis

The points where the graph of a quadratic function meets the $x$-axis are the points with $y = 0$. There the value of the function is $0$, so the $x$-coordinates of these points are found by solving a quadratic equation. Take $y = x^2 - x - 6$.

The intersections with the $x$-axis are where $y = 0$, that is $x^2 - x - 6 = 0$. The left side factors as $(x - 3)(x + 2)$, so $x = 3$ or $x = -2$, and the intersection points are $(3, 0)$ and $(-2, 0)$.

So the $x$-coordinates where a parabola meets the $x$-axis are exactly the solutions of the quadratic equation $ax^2 + bx + c = 0$. The number of intersections equals the number of solutions, decided by the sign of the discriminant $D = b^2 - 4ac$.

• $D > 0$: two distinct intersection points
• $D = 0$: one point of tangency (a repeated root)
• $D < 0$: no intersection (no real solution)

Here $D = (-1)^2 - 4 \cdot 1 \cdot (-6) = 1 + 24 = 25 > 0$, so there are two intersections. The large dots on the graph are those two points, $(3, 0)$ and $(-2, 0)$.