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The given equation is:

\[
[a]^2 - 3[a] + 2 = 0
\]

where \([a]\) denotes the greatest integer function (also known as the floor function, which takes the greatest integer less than or equal to \(a\)).

### Step 1: Solve the quadratic equation

\[
[a]^2 - 3[a] + 2 = 0
\]

This can be factored as:

\[
([a] - 1)([a] - 2) = 0
\]

Therefore, \([a] = 1\) or \([a] = 2\).

### Step 2: Determine the value of \(a\)

Since \([a] = 1\) or \([a] = 2\), this means that \(a\) must lie in the intervals:
- For \([a] = 1\), we have \(1 \leq a < 2\).
- For \([a] = 2\), we have \(2 \leq a < 3\).

### Step 3: Find the maximum integral value of \(10a\)

We now want to maximize the value of \(10a\) for \(a\) in the intervals \(1 \leq a < 2\) and \(2 \leq a < 3\).

- In the interval \(1 \leq a < 2\), the maximum value of \(a\) is just below 2, so the maximum value of \(10a\) is just below \(10 \times 2 = 20\). The greatest integer value here is \(19\).
- In the interval \(2 \leq a < 3\), the maximum value of \(a\) is just below 3, so the maximum value of \(10a\) is just below \(10 \times 3 = 30\). The greatest integer value here is \(29\).

### Conclusion:

The maximum integral value of \(10a\) is \(29\).

---

Would you like further details or have any additional questions?

### Related Questions:
1. How does the greatest integer (floor) function affect solving such equations?
2. What other methods can be used to solve quadratic equations like \([a]^2 - 3[a] + 2 = 0\)?
3. Can the floor function have applications in real-world problems involving measurements?
4. How would the result change if the greatest integer function was replaced by the ceiling function?
5. What are the general rules for maximizing expressions involving floor functions?

### Tip:
Always remember to carefully analyze intervals when working with the greatest integer (floor) function to ensure the solution remains within the correct bounds.

If [a]^2 - 3[a] + 2 = 0, where [a] denotes the greatest integer function, then the maximum integral value of 10a is?

Explore how to solve the quadratic equation [a]^2 - 3[a] + 2 involving the greatest integer function (floor function). Detailed steps to find the maximum integral value of 10a using factored form and interval analysis. Suitable for students in Grades 9-11.

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