Sum of all the solution of the equation `([x])/([x-2])-([x-2])/([x])=(8{x}+12)/([x-2][x])` is (where{*} denotes greatest integer function and {*} represent fractional part function)

Answer

To solve the equation

\[ \frac{[x]}{[x-2]} - \frac{[x-2]}{[x]} = \frac{8\{x\} + 12}{[x-2][x]} \]

where \([x]\) denotes the greatest integer function and \(\{x\}\) denotes the fractional part function, we will follow these steps:

### Step 1: Rewrite the equation

We start by rewriting the equation in a more manageable form. We can express the left-hand side with a common denominator:

\[ \frac{[x]^2 - [x-2]^2}{[x][x-2]} = \frac{8\{x\} + 12}{[x-2][x]} \]

### Step 2: Simplify the left-hand side

The left-hand side can be simplified using the difference of squares:

\[ [x]^2 - [x-2]^2 = ([x] - [x-2])([x] + [x-2]) \]

Since \([x-2] = [x] - 2\), we have:

\[ [x] - [x-2] = 2 \]

Thus, the left-hand side becomes:

\[ \frac{2([x] + [x-2])}{[x][x-2]} = \frac{2([x] + ([x] - 2))}{[x][x-2]} = \frac{2(2[x] - 2)}{[x][x-2]} = \frac{4([x] - 1)}{[x][x-2]} \]

### Step 3: Set the equation

Now we can set the two sides equal to each other:

\[ \frac{4([x] - 1)}{[x][x-2]} = \frac{8\{x\} + 12}{[x-2][x]} \]

### Step 4: Cross-multiply

Cross-multiplying gives us:

\[ 4([x] - 1) \cdot [x-2][x] = (8\{x\} + 12) \cdot [x][x-2] \]

### Step 5: Simplify further

Since both sides have \([x][x-2]\), we can cancel it out, leading to:

\[ 4([x] - 1) = 8\{x\} + 12 \]

### Step 6: Rearranging the equation

Rearranging gives:

\[ 8\{x\} = 4[x] - 4 - 12 \]
\[ 8\{x\} = 4[x] - 16 \]
\[ \{x\} = \frac{4[x] - 16}{8} = \frac{[x] - 4}{2} \]

### Step 7: Analyzing the fractional part

Since \(\{x\}\) must be in the interval \([0, 1)\), we can set up the inequalities:

\[ 0 \leq \frac{[x] - 4}{2} < 1 \]

### Step 8: Solve the inequalities

From \(0 \leq \frac{[x] - 4}{2}\):

\[ [x] - 4 \geq 0 \implies [x] \geq 4 \]

From \(\frac{[x] - 4}{2} < 1\):

\[ [x] - 4 < 2 \implies [x] < 6 \]

Thus, we have:

\[ 4 \leq [x] < 6 \]

### Step 9: Possible values for \([x]\)

The possible integer values for \([x]\) are 4 and 5.

### Step 10: Finding corresponding \(x\) values

1. If \([x] = 4\):
\[ \{x\} = \frac{4 - 4}{2} = 0 \implies x = 4 \]

2. If \([x] = 5\):
\[ \{x\} = \frac{5 - 4}{2} = \frac{1}{2} \implies x = 5.5 \]

### Step 11: Sum of all solutions

The solutions are \(x = 4\) and \(x = 5.5\). The sum of all solutions is:

\[ 4 + 5.5 = 9.5 \]

### Final Answer

The sum of all solutions is:

\[ \boxed{9.5} \]