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To solve the problem using the completing the square method, let's start with the given quadratic equation for the pebble's flight:

\[
y = -8x^2 + 6x - 32.5
\]

### Step 1: Rearrange the equation

We need to rewrite the quadratic equation in the form:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola. Let's start by isolating the quadratic and linear terms:

\[
y = -8x^2 + 6x - 32.5
\]

### Step 2: Factor out the coefficient of \(x^2\)

Factor out \(-8\) from the first two terms:

\[
y = -8(x^2 - \frac{6}{8}x) - 32.5
\]

Simplify the fraction:

\[
y = -8\left(x^2 - \frac{3}{4}x\right) - 32.5
\]

### Step 3: Complete the square

To complete the square, take half of the coefficient of \(x\), square it, and add and subtract it inside the parenthesis:

\[
\text{Half of } -\frac{3}{4} = -\frac{3}{8}, \quad \left(-\frac{3}{8}\right)^2 = \frac{9}{64}
\]

Add and subtract \(\frac{9}{64}\) inside the parenthesis:

\[
y = -8 \left( x^2 - \frac{3}{4}x + \frac{9}{64} - \frac{9}{64} \right) - 32.5
\]

Rewrite it:

\[
y = -8 \left( \left(x - \frac{3}{8}\right)^2 - \frac{9}{64} \right) - 32.5
\]

Distribute the \(-8\):

\[
y = -8 \left(x - \frac{3}{8}\right)^2 + \frac{9}{8} - 32.5
\]

Convert \(-32.5\) to a fraction with a common denominator:

\[
y = -8 \left(x - \frac{3}{8}\right)^2 + \frac{9}{8} - \frac{260}{8}
\]

Combine the fractions:

\[
y = -8 \left(x - \frac{3}{8}\right)^2 - \frac{251}{8}
\]

### Step 4: Analyze the result

The vertex form of the equation is:

\[
y = -8 \left(x - \frac{3}{8}\right)^2 - \frac{251}{8}
\]

The maximum height occurs at \(x = \frac{3}{8}\), but the maximum height itself is \(-\frac{251}{8}\) feet, which is below 0 (the window).

Thus, the pebble does not reach the window. The correct answer is **D**: "It does not reach the window. Its maximum height will be 5 feet below the window."

### Would you like more details, or have any other questions?

#### 5 Questions to Explore:
1. What is the vertex form of a quadratic equation?
2. How do we find the vertex of a parabola?
3. Why is completing the square useful in solving quadratic equations?
4. What does the negative coefficient of \(x^2\) indicate about the parabola's direction?
5. How can we check if the calculated maximum height is correct?

#### Tip:
Completing the square is a valuable technique not only for finding the vertex of a parabola but also for converting quadratic equations into forms suitable for integration and solving optimization problems.

Reggie is trying to toss a pebble at his friend's window to get their attention. The flight of the pebble can be modeled with the equation: y = -8x^2 + 6x - 32.5, where x is time in seconds and y is the height to the window in feet. Use the completing the square process to determine how long it will take for the pebble to reach the window.

Learn how to solve a real-world problem involving the flight of a pebble using the quadratic equation y = -8x^2 + 6x - 32.5. This problem covers key algebraic concepts like completing the square and analyzing parabolas, ideal for students in Grades 9-11.

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