A quadratic is of degree 2.  If the 2nd differences are all the
same then it is quadratic form.   

Write the first and second differences by

To get the first differences:

1. List the numbers in a column
2. For each number in the column, subtract it from the number just
   below it and write it to the right of that number.  

To get the second derivative, do the same with the column of first 
differences

 1  2  12
 3 14 -10
17  4  24
21 28
49


The second differences are not all the same, so it is not a quadratic form.

Here is another set of values that ARE in quadratic form for your comparison:

If the numbers were 1, 6, 15, 28, 45, we would make the first and second differences:

 1  5  4
 6  9  4
15 13  4
28 17 
45

And in this case the second differences are all the same, 4, so it is in
quadratic form.

Then we would substitute in (1,1), (2,6), (3,15) in

y = Ax²+Bx+C

Get three equations in A, B, and C and solve for A, B and C, getting

A=2, B=-1, C=0  and the equation 

y = Ax²+Bx+C

would become

y = 2x²-x

Edwin