Let's start with the given equation:
[tex]\[ x + \frac{1}{x} = 11 \][/tex]
We need to find the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex].
First, square both sides of the given equation:
[tex]\[ \left( x + \frac{1}{x} \right)^2 = 11^2 \][/tex]
Expanding the left-hand side, we get:
[tex]\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 121 \][/tex]
Since [tex]\( x \cdot \frac{1}{x} = 1 \)[/tex], the equation simplifies to:
[tex]\[ x^2 + 2 + \frac{1}{x^2} = 121 \][/tex]
Now, subtract 2 from both sides to isolate [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
[tex]\[ x^2 + \frac{1}{x^2} = 121 - 2 \][/tex]
[tex]\[ x^2 + \frac{1}{x^2} = 119 \][/tex]
Therefore, the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex] is:
[tex]\[ 119 \][/tex]
