TITLE: Recovering a quadratic polynomial from three values using calculus
QUESTION [27 upvotes]: I'm asked to solve this using calculus:

Let $$ f(x) = ax^2 + bx +c .$$ If $ f(1) = 3 $,  $f(2) = 7$, $f(3) = 13$, then find $a$, $b$, and $f(0)$.

I know I can solve this using solving three equations simultaneously. And I can also solve this using Gauss Jordan or Gaussian elimination method by writing the augmented matrix.  But I'm wondering is there any other method to solve this. 
Solving by any method it turns out that $a = b = c = 1$.

REPLY [2 votes]: This caught my eye:
$$
\begin{eqnarray}
f(1)=&3&=2\cdot1+1 \\
f(2)=&7&=3\cdot2+1 \\
f(3)=&13&=4\cdot3+1 \\
\end{eqnarray}
$$
So, $g(x)=f(x)-((x+1)x+1)$ is a quadratic polynomial that has $3$ zeros and so must be the zero polynomial.
Therefore, $f(x)=(x+1)x+1=x^2+x+1$.