If x =` 5-2sqrt(6)` find : ` x^(2)+ (1)/(x^(2))`

Answer

To solve the problem of finding \( x^2 + \frac{1}{x^2} \) where \( x = 5 - 2\sqrt{6} \), we can follow these steps:

### Step 1: Find \( \frac{1}{x} \)
Given \( x = 5 - 2\sqrt{6} \), we need to find \( \frac{1}{x} \).

\[ \frac{1}{x} = \frac{1}{5 - 2\sqrt{6}} \]

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is \( 5 + 2\sqrt{6} \):

\[ \frac{1}{x} = \frac{1 \cdot (5 + 2\sqrt{6})}{(5 - 2\sqrt{6})(5 + 2\sqrt{6})} \]

### Step 2: Simplify the Denominator
Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \):

\[ (5 - 2\sqrt{6})(5 + 2\sqrt{6}) = 5^2 - (2\sqrt{6})^2 = 25 - 24 = 1 \]

Thus, we have:

\[ \frac{1}{x} = 5 + 2\sqrt{6} \]

### Step 3: Calculate \( x + \frac{1}{x} \)
Now we can find \( x + \frac{1}{x} \):

\[ x + \frac{1}{x} = (5 - 2\sqrt{6}) + (5 + 2\sqrt{6}) = 5 + 5 + (-2\sqrt{6} + 2\sqrt{6}) = 10 \]

### Step 4: Use the Identity to Find \( x^2 + \frac{1}{x^2} \)
We know that:

\[ x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 \]

Substituting \( x + \frac{1}{x} = 10 \):

\[ x^2 + \frac{1}{x^2} = 10^2 - 2 = 100 - 2 = 98 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x^2} \) is:

\[ \boxed{98} \]