Find the number of positive integers x satisfying the equation `1/x + 1/(x+1) + 1/(x+2) = 13/12`.

Answer

To solve the equation

\[ \frac{1}{x} + \frac{1}{x+1} + \frac{1}{x+2} = \frac{13}{12}, \]

we will follow these steps:

### Step 1: Combine the fractions on the left side

The left side can be combined into a single fraction. The common denominator for \(x\), \(x+1\), and \(x+2\) is \(x(x+1)(x+2)\). Therefore, we rewrite the left side:

\[ \frac{(x+1)(x+2) + x(x+2) + x(x+1)}{x(x+1)(x+2)}. \]

### Step 2: Expand the numerator

Now we will expand the numerator:

1. \((x+1)(x+2) = x^2 + 3x + 2\),
2. \(x(x+2) = x^2 + 2x\),
3. \(x(x+1) = x^2 + x\).

Adding these together:

\[ x^2 + 3x + 2 + x^2 + 2x + x^2 + x = 3x^2 + 6x + 2. \]

Thus, the left side becomes:

\[ \frac{3x^2 + 6x + 2}{x(x+1)(x+2)}. \]

### Step 3: Set the equation

Now we set the combined fraction equal to the right side:

\[ \frac{3x^2 + 6x + 2}{x(x+1)(x+2)} = \frac{13}{12}. \]

### Step 4: Cross-multiply

Cross-multiplying gives us:

\[ 12(3x^2 + 6x + 2) = 13x(x+1)(x+2). \]

### Step 5: Expand both sides

Expanding the left side:

\[ 36x^2 + 72x + 24. \]

Expanding the right side:

\[ 13x(x^2 + 3x + 2) = 13x^3 + 39x^2 + 26x. \]

### Step 6: Rearrange the equation

Now we rearrange the equation:

\[ 36x^2 + 72x + 24 = 13x^3 + 39x^2 + 26x. \]

Bringing all terms to one side:

\[ 0 = 13x^3 + 39x^2 + 26x - 36x^2 - 72x - 24. \]

Combining like terms gives:

\[ 13x^3 + 3x^2 - 46x - 24 = 0. \]

### Step 7: Solve the cubic equation

Now we will use the Rational Root Theorem to test for possible rational roots. Testing \(x = 2\):

\[ 13(2)^3 + 3(2)^2 - 46(2) - 24 = 13(8) + 3(4) - 92 - 24 = 104 + 12 - 92 - 24 = 0. \]

So \(x = 2\) is a root.

### Step 8: Factor the cubic polynomial

Now we can factor out \(x - 2\) from \(13x^3 + 3x^2 - 46x - 24\). Using synthetic division:

\[ \begin{array}{r|rrrr} 2 & 13 & 3 & -46 & -24 \\ & & 26 & 58 & 24 \\ \hline & 13 & 29 & 12 & 0 \\ \end{array} \]

This gives us:

\[ 13x^2 + 29x + 12 = 0. \]

### Step 9: Solve the quadratic equation

Now we can apply the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-29 \pm \sqrt{29^2 - 4 \cdot 13 \cdot 12}}{2 \cdot 13}. \]

Calculating the discriminant:

\[ 29^2 - 4 \cdot 13 \cdot 12 = 841 - 624 = 217. \]

Thus,

\[ x = \frac{-29 \pm \sqrt{217}}{26}. \]

Since \(\sqrt{217}\) is not a perfect square, we will only consider the integer solution \(x = 2\).

### Step 10: Check for positive integer solutions

The only positive integer solution we found is \(x = 2\).

### Final Answer

Thus, the number of positive integers \(x\) satisfying the equation is:

\[ \boxed{1}. \]