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 their $x$-coordinates are found by solving a quadratic equation¹. Take $y = x^2 - x - 6$.

Factoring

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

The quadratic formula

When the expression does not factor, the formula does the job.

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

Here $D = 1 + 24 = 25 > 0$, so there are two intersections.

Checking with the coefficients

For $ax^2 + bx + c = 0$ the sum of the roots is $-\dfrac{b}{a}$ and their product is $\dfrac{c}{a}$².

The axis and the vertex

Knowing the two intersections nearly fixes the shape of the parabola. The axis passes exactly midway between them, at $\dfrac{3 + (-2)}{2} = \dfrac{1}{2}$, which agrees with the vertex formula $-\dfrac{b}{2a}$. Since a parabola is symmetric about its axis, the two points where it cuts the $x$-axis must be equally far from it. Substituting back gives the vertex $\left(\dfrac{1}{2}, -\dfrac{25}{4}\right)$.

An upward parabola whose vertex lies below the $x$-axis, and a parabola that cuts the axis twice, are two descriptions of the same fact.

Where the sign flips

That is why solving a quadratic inequality begins by finding these points: they are the boundaries of the sign. The large dots on the graph are those two points.

1. Quadratic equation, Wikipedia
2. Vieta's formulas, Wikipedia