Answer
$f(x) = x^{2} + 2x + 5$
Work Step by Step
We must first solve for the functions in terms of a, b, and c:
f(1) = a + b + c = 8
f(2) = 4a + 2b + c = 13
f(3) = 9a + 3b + c = 20
We can then form the matrix, and use Gaussian elimination and back-substitution to solve the matrix:
$\begin{bmatrix}
1 & 1 & 1 & |8\\
4 & 2 & 1 & |13\\
9 & 3 & 1 & |20\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |1\\
0 & -2 & -3 & |-19\\
0 & -6 & -8 & |-52\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |8\\
0 & 2 & 3 & |19\\
0 & -6 & -8 & |-52\\
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |8\\
0 & 2 & 3 & |19\\
0 & 0 & 1 & |5\\
\end{bmatrix}$
C:
c = 5
B:
2b + 3(5) = 19
2b + 15 = 19
2b = 4
b = 2
A:
a + 2 + 5 = 8
a + 7 = 8
a = 1
Using the solution above, the quadratic function is:
$f(x) = x^{2} + 2x + 5$
