### Theory:

A polynomial of the form $$ax^2 + bx + c$$, $$a \neq 0$$ is a quadratic polynomial.

The graph of a quadratic polynomial is a parabolic curve either open upwards or open downwards depending on whether $$a > 0$$ or $$a < 0$$.

Consider the graph of $$y = 2x^2 + 3x + 1$$.

 $$x$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$2x^2$$ $$2(-2)^2$$ $$=$$ $$2(4)$$ $$=$$ $$8$$ $$2(-1)^2$$ $$=$$ $$2(1)$$ $$=$$ $$2$$ $$2(0)^2$$ $$=$$ $$2(0)$$ $$=$$ $$0$$ $$2(1)^2$$ $$=$$ $$2(1)$$ $$=$$ $$2$$ $$2(2)^2$$ $$=$$ $$2(4)$$ $$=$$ $$8$$ $$3x$$ $$3 \times -2$$ $$=$$ $$-6$$ $$3 \times -1$$ $$=$$ $$-3$$ $$3 \times 0$$ $$=$$ $$0$$ $$3 \times 1$$ $$=$$ $$3$$ $$3 \times 2$$ $$=$$ $$6$$ $$2x^2 + 3x + 1$$ $$8$$ $$-$$ $$6$$ $$+$$ $$1$$ $$=$$ $$3$$ $$2$$ $$-$$ $$3$$ $$+$$ $$1$$ $$=$$ $$0$$ $$0$$ $$+$$ $$0$$ $$+$$ $$1$$ $$=$$ $$1$$ $$2$$ $$+$$ $$3$$ $$+$$ $$1$$ $$=$$ $$6$$ $$8$$ $$+$$ $$6$$ $$+$$ $$1$$ $$=$$ $$15$$ $$y$$ $$=$$ $$2x^2 + 3x + 1$$ $$3$$ $$0$$ $$1$$ $$6$$ $$15$$

Join the coordinates $$(-2, 3)$$, $$(-1, 0)$$, $$(0, 1)$$, $$(1, 6)$$ and $$(2, 15)$$ by a smooth curve so as to obtain the graph of $$y = 2x^2 + 3x + 1$$. It is observed that, the graph of the polynomial $$y = 2x^2 + 3x + 1$$ intersects the $$x$$ $$-$$ axis at the points $$(-1, 0)$$ and $$(-0.5, 0)$$.

Thus, we can say that the zero of a quadratic polynomial is the $$x$$ $$-$$ coordinates of the point where the graph of the polynomial intersects the $$x$$ $$-$$ axis.

Let us discuss few cases of graphs of a quadratic polynomial.

Case 1: The graph of the quadratic polynomial cuts the $$x$$ $$-$$ axis at two points, as shown below.  In this case, the number of zeroes of the polynomial is $$2$$.

Case 2: The graph of the quadratic polynomial cuts the $$x$$ $$-$$ axis at one point, as shown below.  In this case, the number of zeroes of the polynomial is $$1$$.

Case 3: The graph of the quadratic polynomial does not intersect the $$x$$ $$-$$ axis at any point, as shown below.  In this case, the number of zeroes of the polynomial is $$0$$.
The quadratic polynomial $$ax^2 + bx + c$$, $$a \neq 0$$, has at most two zeroes which is the $$x$$ $$-$$ coordinates of the point where the graph of the polynomial intersects the $$x$$ $$-$$ axis.