Answer :
Answer:
To solve for \( x \) and \( y \) and then find \( x^2 - y^2 \), let's first simplify \( x \) and \( y \):
Given:
\[ x = \frac{2 - \sqrt{5}}{2 + \sqrt{5}} \]
To eliminate the square root from the denominator, multiply numerator and denominator by the conjugate of the denominator:
\[ x = \frac{(2 - \sqrt{5})(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})} \]
\[ x = \frac{4 - 4\sqrt{5} + 5}{4 - 5} \]
\[ x = \frac{9 - 4\sqrt{5}}{-1} \]
\[ x = - (9 - 4\sqrt{5}) \]
\[ x = 4\sqrt{5} - 9 \]
Next, calculate \( y \):
Given:
\[ y = \frac{2 + \sqrt{5}}{2 - \sqrt{5}} \]
Similarly, multiply numerator and denominator by the conjugate of the denominator:
\[ y = \frac{(2 + \sqrt{5})(2 + \sqrt{5})}{(2 - \sqrt{5})(2 + \sqrt{5})} \]
\[ y = \frac{4 + 4\sqrt{5} + 5}{4 - 5} \]
\[ y = \frac{9 + 4\sqrt{5}}{-1} \]
\[ y = - (9 + 4\sqrt{5}) \]
\[ y = -9 - 4\sqrt{5} \]
Now, find \( x^2 - y^2 \):
\[ x^2 = (4\sqrt{5} - 9)^2 \]
\[ x^2 = 16 \cdot 5 - 72\sqrt{5} + 81 \]
\[ x^2 = 80 - 72\sqrt{5} + 81 \]
\[ x^2 = 161 - 72\sqrt{5} \]
\[ y^2 = (-9 - 4\sqrt{5})^2 \]
\[ y^2 = 81 + 72\sqrt{5} + 16 \]
\[ y^2 = 97 + 72\sqrt{5} \]
Now, calculate \( x^2 - y^2 \):
\[ x^2 - y^2 = (161 - 72\sqrt{5}) - (97 + 72\sqrt{5}) \]
\[ x^2 - y^2 = 161 - 97 - 72\sqrt{5} - 72\sqrt{5} \]
\[ x^2 - y^2 = 64 - 144\sqrt{5} \]
Therefore, the value of \( x^2 - y^2 \) is \( \boxed{64 - 144\sqrt{5}} \).