Explicit formula for a quadratic equation/sequence/pattern

For me, this question is rather fuzzy; if you know at least three data points and you want to know the explicit equation, you can use this procedure:

 

Assuming that the points that you know are (1,5), (2,9), and (3,19):

 

The form of the equation is:  Ax² + Bx + C  =  y

(1,5)  --->   x =1, y = 5   --->   A(1)² + B(1) + C  =  5        --->     A +   B + C  =  5

(2,9)  --->   x = 2, y = 9  --->   A(2)² + B(2) + C  =  9        --->   4A + 2B + C  =  9

(3,19) -->   x = 3, y = 19 -->   A(3)² + B(3) + C  =  19      --->    9A + 3B + C  =  19

 

Combining the first two equations by subtracting the first equation from the second:   3A + B  =  4

Combining the last two equations by subtracting the second equation from the third:  5A + B  =  10

 

Now, using those two equations, subtract the first from the second:                            2A  =  6   --->   A = 3

 

Replacing this value back into the equation:  3A + B  =  4   --->   3(3) + B  =  4   --->   9 + B  =  4   --->   B  =  -5

 

Finally, replace these two values into the equation:  A + B + C  =  5   --->   3 + -5 + C  =  5   --->   C  =  7

 

So, the quadratic equation is:  y  =  3x² - 5x + 7

 

You will need three data points to find a quadratic equation.