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The given expression is:

\[
\frac{x^2 - 14x + 49}{6x^2 - 42x}
\]

Let's simplify this step by step:

1. **Simplify the numerator:**
   \[
   x^2 - 14x + 49
   \]
   This is a perfect square trinomial and can be factored as:
   \[
   (x - 7)^2
   \]

2. **Simplify the denominator:**
   \[
   6x^2 - 42x
   \]
   Factor out the greatest common factor (GCF), which is 6x:
   \[
   6x(x - 7)
   \]

3. **Simplify the entire expression:**
   Now, the expression becomes:
   \[
   \frac{(x - 7)^2}{6x(x - 7)}
   \]
   Cancel out one factor of \( (x - 7) \):
   \[
   \frac{x - 7}{6x}
   \]

Thus, the simplified form of the expression is:
\[
\frac{x - 7}{6x}
\]

Would you like more details or have any questions?

Here are some related questions you might consider:
1. How can you identify perfect square trinomials?
2. What steps should be followed when factoring quadratic expressions?
3. What is the role of the greatest common factor (GCF) in simplifying expressions?
4. How does canceling common factors work in rational expressions?
5. What happens if \(x = 7\) in this expression?

**Tip:** Always check if the expression has any restrictions, like when the denominator is zero, to avoid undefined expressions.

Simplify the rational expression: \(\frac{x^2 - 14x + 49}{6x^2 - 42x}\)

Learn how to simplify the rational expression \(\frac{x^2 - 14x + 49}{6x^2 - 42x}\) using factoring techniques. This problem covers algebraic concepts such as perfect square trinomials, factoring, and simplifying rational expressions. Suitable for students in Grades 9-10.

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