If x `= (1)/(x- 5) , ` find:
` x^(2) + (1)/(x^(2))`

Answer

To solve the equation \( x = \frac{1}{x - 5} \) and find \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x = \frac{1}{x - 5} \]

### Step 2: Square both sides
To eliminate the fraction, we square both sides:
\[ x^2 = \left(\frac{1}{x - 5}\right)^2 \]
This simplifies to:
\[ x^2 = \frac{1}{(x - 5)^2} \]

### Step 3: Rewrite the equation
We can rewrite the equation as:
\[ x^2 (x - 5)^2 = 1 \]

### Step 4: Expand the left side
Expanding the left side gives:
\[ x^2 (x^2 - 10x + 25) = 1 \]
This simplifies to:
\[ x^4 - 10x^3 + 25x^2 - 1 = 0 \]

### Step 5: Find \( \frac{1}{x^2} \)
From \( x^2 = \frac{1}{(x - 5)^2} \), we can find \( \frac{1}{x^2} \):
\[ \frac{1}{x^2} = (x - 5)^2 \]
Expanding this gives:
\[ \frac{1}{x^2} = x^2 - 10x + 25 \]

### Step 6: Add \( x^2 + \frac{1}{x^2} \)
Now we can find \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = x^2 + (x^2 - 10x + 25) \]
This simplifies to:
\[ x^2 + \frac{1}{x^2} = 2x^2 - 10x + 25 \]

### Step 7: Substitute \( x^2 \)
We already have \( x^2 = \frac{1}{(x - 5)^2} \). Substitute this into the equation:
\[ x^2 + \frac{1}{x^2} = 2\left(\frac{1}{(x - 5)^2}\right) - 10x + 25 \]

### Final Result
Thus, the expression \( x^2 + \frac{1}{x^2} \) can be evaluated based on the value of \( x \) derived from the original equation.