12.25 N 0.0228 N2, is part of the quadratic function.
To this a constant of integration must be added, which is
equivalent to the intercept or the value of Y when N is
zero. When a zero-level treatment is included in the data
set, and there are only three levels of input, the average
yield at the zero level can be used as C, the constant of
integration. In this case the average yield of the zero
treatment for IR-22 rice in the trial is 5350 kg/ha, so
this becomes the value of C. The complete quadratic
equation, then, is:

 Y = 5350 + 12.25 N 0.0228 N2

Later, using more orthodox statistical procedures, we will
show that this is the same equation that can be derived by
the method of least squares.
 If no zero treatment is included in the data set, it
is still possible to find the value of C, the constant of
integration. This is done once again by using known
values of Y and N and solving for the unknown constant of
integration. For example, in this case, for a value of N
= 150 the average value of Y = 6675 kg/ha. Hence, the
following information is available:

 6675 = C + 12.25 (150) 0.0228 (22,500)

which comes from the values of N = 150 and N2 = 22,500.
Solving this equation for C results in C = 5350.5, which
is essentially the same value that was calculated before.
Another alternative is to use the mean value of N and the
mean value of Y. However, care must be taken to calculate
the mean value of N2 from the original levels of N. For
example, the three levels of N are 0, 150, and 300.
Squaring each of these values gives 0; 22,500; and 90,000.
The average of these three numbers is 37,500.
 The quadratic response surface being used to
represent the data is shown in Fig. VI-5. Note that in
order to make the curve smooth, the equation can be solved
for levels other than those used in the experiment. For
example, a level of 50 kg/ha of N results in an estimated
value of Y for rice of 5905.
 Now that the response surface has been estimated, it
can be used to help analyze the data. First, find the
level of nitrogen which maximizes physical production. As