To solve the equation \( 6 - 7w = \frac{-1}{9 + 8w} \), let's proceed step by step.

### Step 1: Multiply both sides by \( 9 + 8w \) to eliminate the denominator.

\[
(6 - 7w)(9 + 8w) = -1
\]

### Step 2: Expand the left-hand side.

\[
(6)(9 + 8w) - (7w)(9 + 8w) = -1
\]

\[
54 + 48w - 63w - 56w^2 = -1
\]

Simplify the terms:

\[
54 - 15w - 56w^2 = -1
\]

### Step 3: Bring all terms to one side.

\[
56w^2 + 15w + 54 + 1 = 0
\]

\[
56w^2 + 15w + 55 = 0
\]

### Step 4: Solve this quadratic equation using the quadratic formula. The quadratic formula is:

\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 56 \), \( b = 15 \), and \( c = 55 \).

### Step 5: Calculate the discriminant.

\[
\Delta = b^2 - 4ac = 15^2 - 4(56)(55)
\]

\[
\Delta = 225 - 12320 = -12095
\]

### Step 6: Since the discriminant is negative (\( \Delta = -12095 \)), the equation has no real solutions, only complex solutions.

The solution is:

\[
w = \frac{-15 \pm \sqrt{-12095}}{2(56)}
\]

\[
w = \frac{-15 \pm i\sqrt{12095}}{112}
\]

Thus, the two complex solutions are:

\[
w = \frac{-15 + i\sqrt{12095}}{112} \quad \text{and} \quad w = \frac{-15 - i\sqrt{12095}}{112}
\]

Let me know if you'd like further details or clarifications!

---

Here are 5 related questions:

1. What are the properties of quadratic equations with negative discriminants?
2. How do you interpret complex solutions geometrically?
3. What is the quadratic formula, and why is it useful?
4. Can all quadratic equations be solved using factoring?
5. How do you simplify square roots of large numbers?

**Tip:** If the discriminant is negative, the quadratic equation will always yield complex solutions.

Learn how to solve the quadratic equation 6 - 7w = -1 / (9 + 8w), including step-by-step simplification, use of the quadratic formula, and handling complex solutions. Suitable for advanced high school students (Grades 10-12).

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