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To solve this problem, we need to find the value of \( a \) such that the line tangent to the graph of \( f(x) = 6x^3 - 45x^2 + 109x - 54 \) is parallel to the line \( y = x + 0.4 \).

### Steps to solve:
1. **Determine the slope of the line** \( y = x + 0.4 \). Since this is in the form \( y = mx + b \), the slope is \( m = 1 \).

2. **Find the derivative** \( f'(x) \), which represents the slope of the tangent line at any point \( x \) on the curve \( f(x) \). We will then set this equal to 1 (the slope of the line) to find the value of \( a \).

   Let's find \( f'(x) \):
   \[
   f(x) = 6x^3 - 45x^2 + 109x - 54
   \]
   Differentiate \( f(x) \) with respect to \( x \):
   \[
   f'(x) = 18x^2 - 90x + 109
   \]

3. **Set the derivative equal to 1** (since the line tangent must have the same slope as the line \( y = x + 0.4 \)):
   \[
   18x^2 - 90x + 109 = 1
   \]
   Simplifying the equation:
   \[
   18x^2 - 90x + 108 = 0
   \]
   Divide the entire equation by 18:
   \[
   x^2 - 5x + 6 = 0
   \]
   Factor the quadratic equation:
   \[
   (x - 3)(x - 2) = 0
   \]
   So, \( x = 3 \) or \( x = 2 \).

### Conclusion:
The values of \( a \) are \( x = 3 \) and \( x = 2 \).

Let me know if you need further clarification or more details!

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**Related questions:**
1. How do you find the equation of the tangent line at a specific point?
2. How would you verify that the tangent lines at \( x = 2 \) and \( x = 3 \) are indeed parallel to \( y = x + 0.4 \)?
3. Can you explain how the derivative gives the slope of the tangent line?
4. How do you solve a cubic function like \( f(x) = 6x^3 - 45x^2 + 109x - 54 \) for specific values of \( x \)?
5. What happens if a cubic function has multiple tangent lines with the same slope?

**Tip:** When dealing with tangents, always remember that the slope of the tangent line at a point is given by the derivative of the function at that point.

For what value(s) a is the line tangent to the graph of f(x) = 6x^3 - 45x^2 + 109x - 54 at x = a parallel to the line y = x + 0.4?

This problem involves finding the values of a where the tangent line to the cubic function f(x) = 6x^3 - 45x^2 + 109x - 54 is parallel to the line y = x + 0.4. The solution includes calculating the derivative of f(x) and solving a quadratic equation.

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