Let's go through the solution step-by-step, starting with expressing the profit as a function of the number of items sold using quadratic regression.
### Step 1: Perform Quadratic Regression
Given the data points:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 6 & 9 & 14 \\
\hline
P(x) & 77.6 & 93.4 & 99.4 & 58.6 & -70.4 & -481.4 \\
\hline
\end{array}
\][/tex]
We aim to fit these points to a quadratic function of the form [tex]\( P(x) = ax^2 + bx + c \)[/tex].
The resulting quadratic regression yields the coefficients:
[tex]\[
a = -4.9, \quad b = 30.5, \quad c = 52.0
\][/tex]
Thus, the profit can be expressed as:
[tex]\[
P(x) = -4.9x^2 + 30.5x + 52.0
\][/tex]
### Step 2: Estimate the Profit from Selling 8 Items
Now, we use the quadratic function [tex]\( P(x) = -4.9x^2 + 30.5x + 52.0 \)[/tex] to estimate the profit when [tex]\( x = 8 \)[/tex].
[tex]\[
P(8) = -4.9(8)^2 + 30.5(8) + 52.0
\][/tex]
Calculating this:
[tex]\[
\begin{align*}
P(8) &= -4.9 \cdot 64 + 30.5 \cdot 8 + 52 \\
&= -313.6 + 244 + 52 \\
&= -17.6
\end{align*}
\][/tex]
The estimated profit from selling 8 items is:
[tex]\[
\boxed{-17.6} \text{ dollars}
\][/tex]
### Step 3: Estimate the Number of Items to Make a Profit of -287.6 Dollars
We need to find [tex]\( x \)[/tex] such that:
[tex]\[
P(x) = -287.6
\][/tex]
This translates to solving the equation:
[tex]\[
-4.9x^2 + 30.5x + 52.0 = -287.6
\][/tex]
Rewriting it:
[tex]\[
-4.9x^2 + 30.5x + 339.6 = 0
\][/tex]
Using a numerical solver to find the root of this quadratic equation, we get:
[tex]\[
x \approx -5.78
\][/tex]
Estimating the number of items that need to be sold to make a profit of -287.6 dollars, we get:
[tex]\[
\boxed{-5.78}
\][/tex]
Thus, the detailed solution is complete.
