# Chapter 10 - 10.1 - Matrices and Systems of Equations - 10.1 Exercises - Page 712: 87

$f(x) = -x^{2} + x + 1$

#### Work Step by Step

We must first solve for the functions in terms of a, b, and c: f(1) = a + b + c = 1 f(2) = 4a + 2b + c = -1 f(3) = 9a + 3b + c = -5 We can then form the matrix, and use Gaussian elimination and back-substitution to solve the matrix: $\begin{bmatrix} 1 & 1 & 1 & |1\\ 4 & 2 & 1 & |-1\\ 9 & 3 & 1 & |-5\\ \end{bmatrix}$ ~ $\begin{bmatrix} 1 & 1 & 1 & |1\\ 0 & -2 & -3 & |-5\\ 0 & -6 & -8 & |-14\\ \end{bmatrix}$ ~ $\begin{bmatrix} 1 & 1 & 1 & |1\\ 0 & 2 & 3 & |5\\ 0 & -3 & -4 & |-7\\ \end{bmatrix}$ ~ $\begin{bmatrix} 1 & 1 & 1 & |1\\ 0 & 2 & 3 & |5\\ 0 & 0 & 1 & |1\\ \end{bmatrix}$ C: c = 1 B: 2b + 3(1) = 5 2b + 3 = 5 2b = 2 b = 1 A: a + 1 + 1 = 1 a + 2 = 1 a = -1 Using the solution above, the quadratic function is: $f(x) = -x^{2} + x + 1$

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