4/w-2 - 8/w^2-2w = -2 solve using the lcd

To solve the equation:

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

we can start by simplifying and finding a common denominator (LCD). Here's the step-by-step process:

### Step 1: Factor the denominators
- The denominator of the second fraction \( w^2 - 2w \) can be factored:
  \[
  w^2 - 2w = w(w - 2)
  \]
  So the equation becomes:
  \[
  \frac{4}{w - 2} - \frac{8}{w(w - 2)} = -2
  \]

### Step 2: Find the LCD
- The least common denominator (LCD) between \( w - 2 \) and \( w(w - 2) \) is \( w(w - 2) \).

### Step 3: Rewrite the fractions with the LCD
- Rewrite each fraction so that they have the common denominator \( w(w - 2) \):
  \[
  \frac{4}{w - 2} = \frac{4w}{w(w - 2)}
  \]
  Thus, the equation becomes:
  \[
  \frac{4w}{w(w - 2)} - \frac{8}{w(w - 2)} = -2
  \]

### Step 4: Combine the fractions
- Combine the fractions on the left-hand side:
  \[
  \frac{4w - 8}{w(w - 2)} = -2
  \]

### Step 5: Clear the denominator
- Multiply both sides by \( w(w - 2) \) to eliminate the denominator:
  \[
  4w - 8 = -2w(w - 2)
  \]
  
### Step 6: Expand and simplify
- Expand the right-hand side:
  \[
  4w - 8 = -2w^2 + 4w
  \]
  - Subtract \( 4w \) from both sides:
  \[
  -8 = -2w^2
  \]

### Step 7: Solve for \( w \)
- Divide both sides by \( -2 \):
  \[
  4 = w^2
  \]
  - Take the square root of both sides:
  \[
  w = \pm 2
  \]

### Step 8: Check for extraneous solutions
- Recall that the denominator \( w(w - 2) \) cannot be zero, so \( w = 2 \) is not a valid solution (it makes the denominator zero).
- Therefore, the only solution is:
  \[
  w = -2
  \]

### Final Answer:
\[
w = -2
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How can you identify extraneous solutions in rational equations?
2. Why is it important to factor expressions before finding the LCD?
3. What are common mistakes to avoid when solving equations involving rational expressions?
4. How does multiplying both sides by the LCD simplify the equation?
5. Can you apply a similar method to equations with more complex denominators?

**Tip:** Always check for restrictions in the denominator to avoid extraneous solutions!

Learn how to solve the rational equation 4/(w-2) - 8/(w^2 - 2w) = -2 by using the least common denominator (LCD). This step-by-step solution involves factoring, clearing denominators, and solving quadratic equations. Suitable for Grades 9-11.

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