Recovering a quadratic polynomial from three values using calculus 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 $a = b = c = 1$.

A solution that deliberately doesn't use any calculus...
Given: 
\begin{align}
f(1) \colon\quad \phantom{4}a+\phantom{2}b+c &= 3 \tag{1}\\
f(2) \colon\quad  4a+2b+c&=7 \tag{2}\\
f(3) \colon\quad  9a+3b+c&=13 \tag{3} 
\end{align}
Subtract $(1)$ from $(2)$ and $(3)$, respectively:
\begin{align}
3a+\phantom{2}b &= 4 \tag{A}\\
8a+2b &= 10 \tag{B}
\end{align}
Subtract $2(A)$ from $(B)$: $2a=2$, or $a=1$. 
Plug $a$ into $(A)$: $b=4-3a=4-3=1$. 
Plug $a$ and $b$ into $(1)$:  $c=3-a-b=3-1-1= 1$. 
Hence, $f(x)=x^2+x+1$ and $f(0)=1$. 
