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.