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 \(y = 2x^2 + 3x  + 1\).
 
quadratic.png
 
 
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.
 
Q1.pngQ2.png
 
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.
 
Q4.pngQ3.png
 
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.
 
Q5.pngQ6.png
 
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.