To solve the equation \(\frac{3}{u-2} + \frac{4}{u^2-4} = \frac{1}{u+2}\), let's follow these steps:

### Step 1: Factorize the denominator on the left side
Notice that \( u^2 - 4 \) can be factorized as:
\[
u^2 - 4 = (u - 2)(u + 2)
\]
So the equation becomes:
\[
\frac{3}{u-2} + \frac{4}{(u-2)(u+2)} = \frac{1}{u+2}
\]

### Step 2: Combine the fractions on the left side
To combine the fractions on the left side, find a common denominator, which is \((u-2)(u+2)\):
\[
\frac{3(u+2)}{(u-2)(u+2)} + \frac{4}{(u-2)(u+2)} = \frac{1}{u+2}
\]
This simplifies to:
\[
\frac{3(u+2) + 4}{(u-2)(u+2)} = \frac{1}{u+2}
\]

### Step 3: Expand and simplify the numerator
Expand the numerator on the left side:
\[
3(u + 2) + 4 = 3u + 6 + 4 = 3u + 10
\]
So the equation is now:
\[
\frac{3u + 10}{(u-2)(u+2)} = \frac{1}{u+2}
\]

### Step 4: Eliminate the denominator by cross-multiplication
Since the denominators are equal on both sides, we can equate the numerators:
\[
3u + 10 = 1 \cdot (u - 2)
\]

### Step 5: Solve for \( u \)
Now, expand and solve for \( u \):
\[
3u + 10 = u - 2
\]
Subtract \( u \) from both sides:
\[
2u + 10 = -2
\]
Subtract 10 from both sides:
\[
2u = -12
\]
Finally, divide by 2:
\[
u = -6
\]

### Step 6: Check the solution
We must check that \( u = -6 \) does not make any denominator zero in the original equation.

Substituting \( u = -6 \) back into the original equation, we see that none of the denominators become zero. Therefore, \( u = -6 \) is the solution.

### Final Answer:
\[
u = -6
\]

Would you like more details or have any questions? 

Here are 5 related questions to further explore the topic:
1. How would you solve a similar equation if the numerator had higher powers of \( u \)?
2. What if the equation had complex roots? How would that change the solution process?
3. Can you describe the significance of checking solutions in the context of rational equations?
4. How would this process change if the equation was a rational inequality instead?
5. What are the possible types of solutions for equations involving rational expressions?

**Tip:** Always check your solution in the original equation to ensure that it doesn't cause any undefined expressions, such as division by zero.

Learn how to solve the rational equation (3/(u-2)) + (4/(u^2-4)) = (1/(u+2)) step-by-step with detailed explanations. This problem involves rational equations, factoring, and cross-multiplication methods. Suitable for advanced high school students and college-level learners.

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