`(2(n-1))/(3)-(3(n+1))/(4)=(n+3)/(2)`
In the equation above, what is the value of `n^(2)`?

Answer

To solve the equation

\[ \frac{2(n-1)}{3} - \frac{3(n+1)}{4} = \frac{n+3}{2}, \]

we will follow these steps:

### Step 1: Find a common denominator for the left-hand side.

The denominators are 3 and 4. The least common multiple (LCM) of 3 and 4 is 12.

### Step 2: Rewrite each term with the common denominator.

We can rewrite the left-hand side as:

\[ \frac{2(n-1) \cdot 4}{12} - \frac{3(n+1) \cdot 3}{12} = \frac{8(n-1)}{12} - \frac{9(n+1)}{12}. \]

### Step 3: Combine the fractions.

Now we can combine the fractions:

\[ \frac{8(n-1) - 9(n+1)}{12}. \]

### Step 4: Expand the numerators.

Expanding the numerators gives:

\[ 8(n-1) = 8n - 8, \]
\[ 9(n+1) = 9n + 9. \]

So we have:

\[ 8(n-1) - 9(n+1) = 8n - 8 - 9n - 9 = -n - 17. \]

### Step 5: Write the left-hand side as a single fraction.

Thus, the left-hand side becomes:

\[ \frac{-n - 17}{12}. \]

### Step 6: Set the left-hand side equal to the right-hand side.

Now we set this equal to the right-hand side:

\[ \frac{-n - 17}{12} = \frac{n + 3}{2}. \]

### Step 7: Eliminate the fractions by multiplying both sides by 12.

Multiplying both sides by 12 gives:

\[ -n - 17 = 6(n + 3). \]

### Step 8: Expand the right-hand side.

Expanding the right-hand side results in:

\[ -n - 17 = 6n + 18. \]

### Step 9: Rearrange the equation.

Now, we rearrange the equation to isolate \(n\):

\[ -n - 6n = 18 + 17, \]
\[ -7n = 35. \]

### Step 10: Solve for \(n\).

Dividing both sides by -7 gives:

\[ n = -5. \]

### Step 11: Find \(n^2\).

Finally, we calculate \(n^2\):

\[ n^2 = (-5)^2 = 25. \]

Thus, the value of \(n^2\) is

\[ \boxed{25}. \]