Geometrical Meaning of the Zeroes of a Quadratic Polynomial
From our observation earlier about the shape of the graph of $y = ax^2 + bx + c$ , the following three cases can happen:
Case (i) : Here, the graph cuts x-axis at two distinct points A and A′.
The x-coordinates of A and A′ are the two zeroes of the quadratic polynomial
$ax^2 + bx + c$ in this case .
Case (ii) : Here, the graph cuts the x-axis at exactly one point, i.e., at two coincident points. So, the two points A and A′ of Case (i) coincide here to become one point A (see Fig. 2.4).
The x-coordinate of A is the only zero for the quadratic polynomial $ax^2 + bx + c$ in this case.
Case (iii) : Here, the graph is either completely above the x-axis or completely below the x-axis. So, it does not cut the x-axis at any point (see Fig. 2.5).
So, the quadratic polynomial $ax^2 + bx + c$ has no zero in this case.
So, you can see geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero.
This also means that a polynomial of degree 2 has atmost two zeroes.
