https://file.aiquickdraw.com/mathbot/file/1726155222829udmlafx0.jpg

The equation from the image is:

\[
\frac{8w^2}{w^2 - 1} = \frac{4}{w - 1} + \frac{9w}{w + 1}
\]

### Steps to solve:

1. **Factor** the denominator on the left-hand side:
   \[
   w^2 - 1 = (w - 1)(w + 1)
   \]
   So, the equation becomes:
   \[
   \frac{8w^2}{(w - 1)(w + 1)} = \frac{4}{w - 1} + \frac{9w}{w + 1}
   \]

2. **Clear the denominators** by multiplying both sides of the equation by \((w - 1)(w + 1)\) to eliminate fractions:
   \[
   8w^2 = 4(w + 1) + 9w(w - 1)
   \]

3. **Expand** both sides:
   \[
   8w^2 = 4w + 4 + 9w^2 - 9w
   \]
   Simplifying further:
   \[
   8w^2 = 9w^2 - 5w + 4
   \]

4. **Move all terms** to one side:
   \[
   8w^2 - 9w^2 + 5w - 4 = 0
   \]
   Simplify:
   \[
   -w^2 + 5w - 4 = 0
   \]

5. **Multiply through by -1** to make the quadratic easier to solve:
   \[
   w^2 - 5w + 4 = 0
   \]

6. **Factor** the quadratic:
   \[
   (w - 4)(w - 1) = 0
   \]

7. **Solve for \( w \)**:
   \[
   w = 4 \quad \text{or} \quad w = 1
   \]

8. **Check for extraneous solutions**:
   We must exclude values of \( w \) that make any denominators zero. From the original equation, the denominators \( w - 1 \) and \( w + 1 \) are present, so \( w = 1 \) is **not valid** because it makes the denominator zero.

Thus, the only valid solution is:
\[
w = 4
\]

Would you like further details or explanations on any part of the solution?

### Follow-up questions:
1. Can you explain how to factor quadratic equations?
2. How can we identify extraneous solutions in rational equations?
3. What does it mean to "clear the denominators"?
4. How can we check our solutions when solving equations involving fractions?
5. What happens when both solutions of a quadratic equation are extraneous?

**Tip:** Always check for restrictions in rational equations by identifying values that make denominators zero.

Solve for w: 8w^2 / (w^2 - 1) = 4 / (w - 1) + 9w / (w + 1)

Learn how to solve a rational equation involving a quadratic expression: 8w^2 / (w^2 - 1) = 4 / (w - 1) + 9w / (w + 1). Follow a step-by-step solution involving factoring, expanding, and checking for extraneous solutions. Suitable for Grades 9-11.

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