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To find the vertical asymptotes of the rational function:

\[
f(x) = \frac{x^2 - 9x + 14}{x^2 - 5x + 6}
\]

### Steps:
1. **Vertical asymptotes occur where the denominator is zero**, but the numerator is not zero at the same points. So, we first solve for when the denominator equals zero.

The denominator is:

\[
x^2 - 5x + 6 = 0
\]

2. **Solve the quadratic equation**:

\[
x^2 - 5x + 6 = 0
\]

We can factor it:

\[
(x - 2)(x - 3) = 0
\]

So, the denominator is zero when \(x = 2\) and \(x = 3\).

3. **Check if the numerator is zero at these values**:
   The numerator is \(x^2 - 9x + 14\). Substituting \(x = 2\) and \(x = 3\) into the numerator:

   - For \(x = 2\):
     \[
     2^2 - 9(2) + 14 = 4 - 18 + 14 = 0
     \]
   - For \(x = 3\):
     \[
     3^2 - 9(3) + 14 = 9 - 27 + 14 = -4
     \]

   So, at \(x = 2\), the numerator is also zero, meaning there is **no vertical asymptote** at \(x = 2\) (it is a removable discontinuity).

   At \(x = 3\), the numerator is not zero, so there is a **vertical asymptote** at \(x = 3\).

### Conclusion:
The vertical asymptote is at \(x = 3\), which corresponds to the second option: \(x = 3\).

---

Would you like further clarification on any part of this process?

Here are some related questions to explore further:
1. How do horizontal asymptotes differ from vertical asymptotes?
2. What is a removable discontinuity in a function?
3. How do you find the horizontal asymptote for this function?
4. Can there be more than one vertical asymptote for a rational function?
5. What are the steps to factor a quadratic expression efficiently?

**Tip**: Always check if the numerator cancels out any of the roots of the denominator to identify removable discontinuities!

Find the vertical asymptote(s) for the following function: f(x) = (x^2 - 9x + 14) / (x^2 - 5x + 6)

Learn how to find the vertical asymptotes for the rational function f(x) = (x^2 - 9x + 14) / (x^2 - 5x + 6) by solving for the points where the denominator equals zero but the numerator does not. This step-by-step solution demonstrates factoring quadratics and identifying removable discontinuities. Ideal for high school students.

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