Answer :
To find \( (X - 4)^2 + (Y - 3)^2 \), where \( X = 4 + \sqrt{3} + \sqrt{5} \) and \( Y = 3 + \sqrt{3} - \sqrt{5} \), we will proceed with the following steps:
1. Calculate \( X - 4 \):
\[ X - 4 = 4 + \sqrt{3} + \sqrt{5} - 4 = \sqrt{3} + \sqrt{5} \]
2. Calculate \( Y - 3 \):
\[ Y - 3 = 3 + \sqrt{3} - \sqrt{5} - 3 = \sqrt{3} - \sqrt{5} \]
3. Compute \( (X - 4)^2 + (Y - 3)^2 \):
\[ (X - 4)^2 = (\sqrt{3} + \sqrt{5})^2 = (\sqrt{3})^2 + 2 \cdot \sqrt{3} \cdot \sqrt{5} + (\sqrt{5})^2 \]
\[ (X - 4)^2 = 3 + 2\sqrt{15} + 5 = 8 + 2\sqrt{15} \]
\[ (Y - 3)^2 = (\sqrt{3} - \sqrt{5})^2 = (\sqrt{3})^2 - 2 \cdot \sqrt{3} \cdot \sqrt{5} + (\sqrt{5})^2 \]
\[ (Y - 3)^2 = 3 - 2\sqrt{15} + 5 = 8 - 2\sqrt{15} \]
4. Add \( (X - 4)^2 \) and \( (Y - 3)^2 \):
\[ (X - 4)^2 + (Y - 3)^2 = (8 + 2\sqrt{15}) + (8 - 2\sqrt{15}) \]
\[ (X - 4)^2 + (Y - 3)^2 = 16 \]
Therefore, \( (X - 4)^2 + (Y - 3)^2 = \boxed{16} \).
1. Calculate \( X - 4 \):
\[ X - 4 = 4 + \sqrt{3} + \sqrt{5} - 4 = \sqrt{3} + \sqrt{5} \]
2. Calculate \( Y - 3 \):
\[ Y - 3 = 3 + \sqrt{3} - \sqrt{5} - 3 = \sqrt{3} - \sqrt{5} \]
3. Compute \( (X - 4)^2 + (Y - 3)^2 \):
\[ (X - 4)^2 = (\sqrt{3} + \sqrt{5})^2 = (\sqrt{3})^2 + 2 \cdot \sqrt{3} \cdot \sqrt{5} + (\sqrt{5})^2 \]
\[ (X - 4)^2 = 3 + 2\sqrt{15} + 5 = 8 + 2\sqrt{15} \]
\[ (Y - 3)^2 = (\sqrt{3} - \sqrt{5})^2 = (\sqrt{3})^2 - 2 \cdot \sqrt{3} \cdot \sqrt{5} + (\sqrt{5})^2 \]
\[ (Y - 3)^2 = 3 - 2\sqrt{15} + 5 = 8 - 2\sqrt{15} \]
4. Add \( (X - 4)^2 \) and \( (Y - 3)^2 \):
\[ (X - 4)^2 + (Y - 3)^2 = (8 + 2\sqrt{15}) + (8 - 2\sqrt{15}) \]
\[ (X - 4)^2 + (Y - 3)^2 = 16 \]
Therefore, \( (X - 4)^2 + (Y - 3)^2 = \boxed{16} \).