If `m + (1)/( m - 2)` = 4 find the value of `(m - 2) ^(2)+ (1)/( ( m - 2)^(2))`

Options

A. `-2`
B. 0
C. 2
D. 4

Answer

To solve the equation \( m + \frac{1}{m - 2} = 4 \) and find the value of \( (m - 2)^2 + \frac{1}{(m - 2)^2} \), we can follow these steps:

### Step 1: Isolate \( m \)
Start with the given equation:
\[ m + \frac{1}{m - 2} = 4 \]
Subtract \( 4 \) from both sides:
\[ m + \frac{1}{m - 2} - 4 = 0 \]
Rearranging gives:
\[ m - 4 + \frac{1}{m - 2} = 0 \]
This can be rewritten as:
\[ m - 2 + 1 = 4 \]

### Step 2: Rewrite the equation
We can express \( m \) in terms of \( m - 2 \):
\[ m - 2 + \frac{1}{m - 2} = 2 \]
Let \( x = m - 2 \). Then the equation becomes:
\[ x + \frac{1}{x} = 2 \]

### Step 3: Square both sides
Square both sides to eliminate the fraction:
\[ \left(x + \frac{1}{x}\right)^2 = 2^2 \]
This expands to:
\[ x^2 + 2 + \frac{1}{x^2} = 4 \]

### Step 4: Rearrange the equation
Now, rearranging gives:
\[ x^2 + \frac{1}{x^2} = 4 - 2 \]
Thus:
\[ x^2 + \frac{1}{x^2} = 2 \]

### Step 5: Substitute back
Recall that \( x = m - 2 \), so we have:
\[ (m - 2)^2 + \frac{1}{(m - 2)^2} = 2 \]

### Final Answer
Therefore, the value of \( (m - 2)^2 + \frac{1}{(m - 2)^2} \) is:
\[ \boxed{2} \]