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To find the values of \( x \) for which the curve \( y = x^3 - x^2 + 9 \) has a tangent line parallel to the line \( y = 5x + 1 \), we need to understand that parallel lines have the same slope.

### Solution:

1. **Determine the slope of the given line** \( y = 5x + 1 \):
   - The slope of \( y = 5x + 1 \) is \( 5 \), since it’s the coefficient of \( x \).

2. **Find the derivative of the curve \( y = x^3 - x^2 + 9 \)**:
   - To find the slope of the tangent line to the curve, we differentiate \( y = x^3 - x^2 + 9 \) with respect to \( x \):
     \[
     \frac{dy}{dx} = 3x^2 - 2x
     \]

3. **Set the derivative equal to 5** to find the points where the tangent slope is 5:
   \[
   3x^2 - 2x = 5
   \]
   - Rearrange this equation to:
     \[
     3x^2 - 2x - 5 = 0
     \]

4. **Solve the quadratic equation** \( 3x^2 - 2x - 5 = 0 \):
   - Using the quadratic formula:
     \[
     x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-5)}}{2 \cdot 3}
     \]
     \[
     x = \frac{2 \pm \sqrt{4 + 60}}{6}
     \]
     \[
     x = \frac{2 \pm \sqrt{64}}{6}
     \]
     \[
     x = \frac{2 \pm 8}{6}
     \]
     - This gives:
       \[
       x = \frac{10}{6} = \frac{5}{3} \quad \text{and} \quad x = \frac{-6}{6} = -1
       \]

5. **Conclusion**:
   - The values of \( x \) where the curve has a tangent line parallel to \( y = 5x + 1 \) are \( x = \frac{5}{3} \) and \( x = -1 \).

So, the correct answer is:
- **(c) \( x = \frac{5}{3} \) and \( x = -1 \)**.

Would you like further details on any step?

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Here are some related questions to deepen understanding:

1. How do you determine the slope of a tangent line to a curve at a specific point?
2. What is the geometric meaning of setting the derivative equal to the slope of a given line?
3. How would the solution change if the line’s equation was different, such as \( y = 3x + 4 \)?
4. Why is the quadratic formula useful in solving this type of problem?
5. How can you verify that the points \( x = 5/3 \) and \( x = -1 \) actually give a tangent parallel to the line?

**Tip**: When you encounter a curve with a tangent parallel to a line, the first step is always to match the slopes by using the derivative.

At what values of x does the curve y = x^3 - x^2 + 9 have a tangent line parallel to the line y = 5x + 1?

Discover how to determine the values of x for which the tangent line to the curve y = x^3 - x^2 + 9 is parallel to the line y = 5x + 1. This detailed solution covers key concepts in differentiation and quadratic equations, ideal for students in Grades 11-12.

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