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Let's work through this problem.

Given the quadratic equation:
\[
8w^2 + 2w - 15 = 0
\]

We use the quadratic formula to find the values for \(N\), \(M\), and \(D\) as per the form given:
\[
w = \frac{N \pm \sqrt{D}}{M}
\]

The quadratic formula is:
\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this equation:
- \( a = 8 \)
- \( b = 2 \)
- \( c = -15 \)

### Step 1: Calculate \( N \), \( M \), and \( D \)

1. **Calculate the discriminant** \( D = b^2 - 4ac \):
   \[
   D = (2)^2 - 4 \cdot 8 \cdot (-15) = 4 + 480 = 484
   \]

2. **Calculate \( N \)**:
   \[
   N = -b = -2
   \]

3. **Calculate \( M \)**:
   \[
   M = 2a = 2 \cdot 8 = 16
   \]

So we have:
- \( N = -2 \)
- \( D = 484 \)
- \( M = 16 \)

These values match the filled-in answers for part (A).

### Step 2: Simplify the Radical and Solve for \( w \)

Since \( D = 484 \), we find \( \sqrt{D} = \sqrt{484} = 22 \).

Now we can substitute into the formula:
\[
w = \frac{-2 \pm 22}{16}
\]

This gives two solutions:
1. \( w = \frac{-2 + 22}{16} = \frac{20}{16} = \frac{5}{4} \)
2. \( w = \frac{-2 - 22}{16} = \frac{-24}{16} = \frac{-3}{2} \)

### Final Answer for Part (B):
\[
w = \frac{5}{4}, -\frac{3}{2}
\]

---

Would you like more details on each step? 

Here are some related questions for further understanding:

1. How do you determine the values of \( a \), \( b \), and \( c \) in a quadratic equation?
2. Why is the discriminant important in finding the nature of the roots?
3. What does it mean if the discriminant is a perfect square?
4. How would the solution differ if \( D \) were not a perfect square?
5. How can you verify that the answers for \( w \) are correct?

**Tip:** Always check if the discriminant is a perfect square; this helps simplify square roots easily, leading to simplified solutions.

The equation 8w^2 + 2w - 15 = 0 has solutions of the form \( w = \frac{N \pm \sqrt{D}}{M} \). Use the quadratic formula to solve this equation and find the appropriate integer values of N, M, and D. Then, simplify the radical and the resulting solutions.

Learn how to solve the quadratic equation 8w^2 + 2w - 15 using the quadratic formula and simplify the solutions. Find the values for N, M, and D and express the final answer in simplified form. Ideal for students in Grades 9-10 exploring algebra and quadratic equations.

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