Let `x=(5+2sqrt6)^(n),ninN`, then find the value of `x-x^(2)+x[x]`, where [.] denotes greatest integer function.

Answer

To solve the problem step by step, we start with the expression given in the question.

### Step 1: Define the variables
Let \( x = (5 + 2\sqrt{6})^n \), where \( n \in \mathbb{N} \).

### Step 2: Introduce another variable
Let \( y = (5 - 2\sqrt{6})^n \). Note that \( 5 - 2\sqrt{6} \) is a small positive number (approximately 0.1), so \( y \) will be a very small positive number when \( n \) is a natural number.

### Step 3: Calculate \( x + y \)
We can calculate \( x + y \):
\[ x + y = (5 + 2\sqrt{6})^n + (5 - 2\sqrt{6})^n \]
Using the Binomial Theorem, we can expand both terms:
\[ x + y = \sum_{k=0}^{n} \binom{n}{k} 5^{n-k} (2\sqrt{6})^k + \sum_{k=0}^{n} \binom{n}{k} 5^{n-k} (-2\sqrt{6})^k \]
The odd powers of \( 2\sqrt{6} \) will cancel out, leaving:
\[ x + y = 2 \sum_{k=0, \text{ even}}^{n} \binom{n}{k} 5^{n-k} (2\sqrt{6})^k \]
This sum is an integer, so we denote it as \( I \).

### Step 4: Calculate \( xy \)
Now, we calculate \( xy \):
\[ xy = (5 + 2\sqrt{6})^n (5 - 2\sqrt{6})^n = ((5)^2 - (2\sqrt{6})^2)^n = (25 - 24)^n = 1^n = 1 \]

### Step 5: Relate \( x \) and \( y \)
From the above, we have:
\[ x + y = I \quad \text{and} \quad xy = 1 \]
This gives us a quadratic equation:
\[ t^2 - It + 1 = 0 \]
where \( t = x \) or \( t = y \).

### Step 6: Use the greatest integer function
The greatest integer function \( [x] \) can be expressed as:
\[ [x] = I - y \quad \text{(since \( y \) is very small)} \]

### Step 7: Calculate \( x - x^2 + x[x] \)
We need to find:
\[ x - x^2 + x[x] \]
Substituting for \( [x] \):
\[ x - x^2 + x(I - y) \]
This simplifies to:
\[ x - x^2 + xI - xy \]
Since \( xy = 1 \):
\[ x - x^2 + xI - 1 \]
Rearranging gives:
\[ x + xI - x^2 - 1 \]

### Step 8: Substitute \( I \)
Recall that \( I = x + y \), thus:
\[ x + x(x + y) - x^2 - 1 = x + x^2 + xy - x^2 - 1 \]
This simplifies to:
\[ x + 1 - 1 = x \]

### Final Result
Thus, the value of \( x - x^2 + x[x] \) is:
\[ \boxed{x} \]