Certainly! Let's break down each step in detail.
### Part A: Setting Up the System of Equations
We are given three points that lie on the parabola: [tex]\((3,8)\)[/tex], [tex]\(\left(5, \frac{20}{3}\right)\)[/tex], and [tex]\((6,5)\)[/tex]. We want to find the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] of the parabola described by the equation [tex]\(y = ax^2 + bx + c\)[/tex].
Plugging each point into the parabola equation yields:
1. For the point [tex]\((3, 8)\)[/tex]:
[tex]\[
8 = 3^2a + 3b + c = 9a + 3b + c
\][/tex]
2. For the point [tex]\((5, \frac{20}{3})\)[/tex]:
[tex]\[
\frac{20}{3} = 5^2a + 5b + c = 25a + 5b + c
\][/tex]
3. For the point [tex]\((6, 5)\)[/tex]:
[tex]\[
5 = 6^2a + 6b + c = 36a + 6b + c
\][/tex]
These equations create the following system of equations:
[tex]\[
\begin{array}{l}
8 = 9a + 3b + c \\
\frac{20}{3} = 25a + 5b + c \\
5 = 36a + 6b + c
\end{array}
\][/tex]
### Part B: Solving Using Matrix Manipulation
We can represent this system of equations in matrix form [tex]\(AX = B\)[/tex] where:
[tex]\[
A = \begin{pmatrix}
9 & 3 & 1 \\
25 & 5 & 1 \\
36 & 6 & 1
\end{pmatrix}, \quad X = \begin{pmatrix}
a \\
b \\
c
\end{pmatrix}, \quad B = \begin{pmatrix}
8 \\
\frac{20}{3} \\
5
\end{pmatrix}
\][/tex]
To solve for [tex]\(X\)[/tex], we need to find [tex]\(A^{-1}\)[/tex] and compute [tex]\(X = A^{-1}B\)[/tex].
Let's find the inverse of [tex]\(A\)[/tex]:
[tex]\[
A^{-1} = \begin{pmatrix}
0.1667 & -0.5 & 0.3333 \\
-1.8333 & 4.5 & -2.6667 \\
5 & -9 & 5
\end{pmatrix}
\][/tex]
Now we multiply [tex]\(A^{-1}\)[/tex] by [tex]\(B\)[/tex] to find [tex]\(X\)[/tex]:
[tex]\[
X = A^{-1}B = \begin{pmatrix}
0.1667 & -0.5 & 0.3333 \\
-1.8333 & 4.5 & -2.6667 \\
5 & -9 & 5
\end{pmatrix} \begin{pmatrix}
8 \\
\frac{20}{3} \\
5
\end{pmatrix} = \begin{pmatrix}
-0.3333 \\
2.0000 \\
5.0000
\end{pmatrix}
\][/tex]
Hence, the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are:
[tex]\[
a = -0.3333, \quad b = 2.0000, \quad c = 5.0000
\][/tex]
### Final Equation of the Parabola
Using the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], the equation of the parabola is:
[tex]\[
y = -0.3333x^2 + 2x + 5
\][/tex]
Thus, the final equation of the parabola passing through the points [tex]\((3,8)\)[/tex], [tex]\(\left(5, \frac{20}{3}\right)\)[/tex], and [tex]\((6,5)\)[/tex] is:
[tex]\[
y = -\frac{1}{3}x^2 + 2x + 5
\][/tex]
