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To find the best-fit curve of the form \( p(x) = ax + bx^2 \) for the data points provided, we can use the least squares method to determine the values of \( a \) and \( b \).

The data points given are:
\[
(0, 0), (1, 5), (2, 1)
\]

### Step 1: Set Up the Equations Using Least Squares
We need to find values of \( a \) and \( b \) that minimize the sum of the squared residuals:
\[
\sum (y_i - (ax_i + bx_i^2))^2
\]

This can be expanded to a system of equations based on each data point. Specifically, we substitute each \((x_i, y_i)\) into \( y = ax + bx^2 \) and create a system of equations.

For our three points:
1. When \( x = 0 \) and \( y = 0 \):
   \[
   0 = a \cdot 0 + b \cdot 0^2 \Rightarrow 0 = 0
   \]

2. When \( x = 1 \) and \( y = 5 \):
   \[
   5 = a \cdot 1 + b \cdot 1^2 \Rightarrow 5 = a + b
   \]

3. When \( x = 2 \) and \( y = 1 \):
   \[
   1 = a \cdot 2 + b \cdot 2^2 \Rightarrow 1 = 2a + 4b
   \]

Thus, we get the following system of linear equations:
\[
\begin{cases}
5 = a + b \\
1 = 2a + 4b
\end{cases}
\]

### Step 2: Solve the System of Equations
We can solve this system by substitution or elimination.

1. From the first equation:
   \[
   a = 5 - b
   \]

2. Substitute \( a = 5 - b \) into the second equation:
   \[
   1 = 2(5 - b) + 4b
   \]
   \[
   1 = 10 - 2b + 4b
   \]
   \[
   1 = 10 + 2b
   \]
   \[
   2b = 1 - 10 = -9
   \]


Find the best fit of the form p(x) = ax + bx^2 to the data (x, y) = (0,0), (1,5), (2,1).

Learn how to find the best-fit quadratic function of the form p(x) = ax + bx^2 for given data points using the least squares method. This solution includes setting up and solving linear equations to find optimal values for parameters a and b.

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