Let's find the value of [tex]\( x^2 + \frac{1}{x^2} \)[/tex] given [tex]\( x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \)[/tex].
1. First, define the expression for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}
\][/tex]
2. Square the expression for [tex]\( x \)[/tex] to find [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \left( \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \right)^2
\][/tex]
Based on calculated results:
[tex]\[
x^2 \approx 61.98386676965928
\][/tex]
3. Calculate the reciprocal of [tex]\( x^2 \)[/tex], which is [tex]\( \frac{1}{x^2} \)[/tex]:
[tex]\[
\frac{1}{x^2} \approx 0.016133230340664932
\][/tex]
4. Add [tex]\( x^2 \)[/tex] and [tex]\( \frac{1}{x^2} \)[/tex]:
[tex]\[
x^2 + \frac{1}{x^2} \approx 61.98386676965928 + 0.016133230340664932
\][/tex]
[tex]\[
x^2 + \frac{1}{x^2} \approx 61.99999999999994
\][/tex]
5. Now, let's match this result to the given options:
[tex]\[
\text{A) 54}
\][/tex]
[tex]\[
\text{B) 14}
\][/tex]
[tex]\[
\text{C) 62}
\][/tex]
[tex]\[
\text{D) 66}
\][/tex]
Based on our calculated values, the closest and the most accurate answer is:
[tex]\[
\boxed{62}
\][/tex]
