To find [tex]\( x^2 + y^2 \)[/tex] where [tex]\( x = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \)[/tex] and [tex]\( y = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \)[/tex], we will proceed as follows:
1. Establish the expressions for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}}
\][/tex]
[tex]\[
y = \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}
\][/tex]
2. Square both [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x^2 = \left( \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} \right)^2
\][/tex]
[tex]\[
y^2 = \left( \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} \right)^2
\][/tex]
3. Simplify [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex]:
Consider [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{(\sqrt{3} + \sqrt{2})^2}{(\sqrt{3} - \sqrt{2})^2}
\][/tex]
The numerator [tex]\((\sqrt{3} + \sqrt{2})^2\)[/tex]:
[tex]\[
(\sqrt{3} + \sqrt{2})^2 = 3 + 2\sqrt{6} + 2
\][/tex]
[tex]\[
= 5 + 2\sqrt{6}
\][/tex]
The denominator [tex]\((\sqrt{3} - \sqrt{2})^2\)[/tex]:
[tex]\[
(\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2
\][/tex]
[tex]\[
= 5 - 2\sqrt{6}
\][/tex]
Thus,
[tex]\[
x^2 = \frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}}
\][/tex]
Now, consider [tex]\( y^2 \)[/tex]:
[tex]\[
y^2 = \frac{(\sqrt{3} - \sqrt{2})^2}{(\sqrt{3} + \sqrt{2})^2}
\][/tex]
The numerator [tex]\((\sqrt{3} - \sqrt{2})^2\)[/tex]:
[tex]\[
(\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2
\][/tex]
[tex]\[
= 5 - 2\sqrt{6}
\][/tex]
The denominator [tex]\((\sqrt{3} + \sqrt{2})^2\)[/tex]:
[tex]\[
(\sqrt{3} + \sqrt{2})^2 = 3 + 2\sqrt{6} + 2
\][/tex]
[tex]\[
= 5 + 2\sqrt{6}
\][/tex]
Thus,
[tex]\[
y^2 = \frac{5 - 2\sqrt{6}}{5 + 2\sqrt{6}}
\][/tex]
4. Add [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex] together:
[tex]\[
x^2 + y^2 = \frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}} + \frac{5 - 2\sqrt{6}}{5 + 2\sqrt{6}}
\][/tex]
5. Express the result in a simple form:
The fractions [tex]\( \frac{5 + 2\sqrt{6}}{5 - 2\sqrt{6}} \)[/tex] and [tex]\( \frac{5 - 2\sqrt{6}}{5 + 2\sqrt{6}} \)[/tex] are reciprocals of each other. When added, their sum can be expressed as:
[tex]\[
x^2 + y^2 = \frac{(5 + 2\sqrt{6})^2 + (5 - 2\sqrt{6})^2}{(5 - 2\sqrt{6})(5 + 2\sqrt{6})}
\][/tex]
Simplifying the denominator:
[tex]\[
(5 - 2\sqrt{6})(5 + 2\sqrt{6}) = 25 - (2\sqrt{6})^2 = 25 - 24 = 1
\][/tex]
Therefore, the sum in the numerator remains unchanged:
[tex]\[
x^2 + y^2 = (5 + 2\sqrt{6})^2 + (5 - 2\sqrt{6})^2
\][/tex]
6. Sum up the numerators:
[tex]\[
(5 + 2\sqrt{6})^2 = 25 + 20\sqrt{6} + 24 = 49 + 20\sqrt{6}
\][/tex]
[tex]\[
(5 - 2\sqrt{6})^2 = 25 - 20\sqrt{6} + 24 = 49 - 20\sqrt{6}
\][/tex]
Adding these:
[tex]\[
(49 + 20\sqrt{6}) + (49 - 20\sqrt{6}) = 49 + 49 = 98
\][/tex]
Thus,
[tex]\[
x^2 + y^2 = \boxed{98}
\][/tex]
