Given the equation:
[tex]\[ x + \frac{1}{x} = 7 \][/tex]
We need to find the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex].
Let's start by squaring both sides of the given equation:
[tex]\[
\left( x + \frac{1}{x} \right)^2 = 7^2
\][/tex]
Expanding the left side:
[tex]\[
\left( x + \frac{1}{x} \right)^2 = x^2 + 2 \cdot \frac{x}{x} + \frac{1}{x^2}
\][/tex]
[tex]\[
x^2 + 2 + \frac{1}{x^2} = 49
\][/tex]
Next, we isolate [tex]\( x^2 + \frac{1}{x^2} \)[/tex]:
[tex]\[
x^2 + \frac{1}{x^2} = 49 - 2
\][/tex]
[tex]\[
x^2 + \frac{1}{x^2} = 47
\][/tex]
Therefore, the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex] is
[tex]\[
\boxed{47}
\][/tex]
