七月是好月 寫到:
Sorry! 是我眼花,把 8 看成 5!
解:
sqrt(16 + sqrt(252))
= sqrt(16 + 2 sqrt(63))
= sqrt( (sqrt(9) + sqrt(7))² )
= 3 + sqrt(7)
sqrt(8 - sqrt(28))
= sqrt(8 - 2 sqrt(7))
= sqrt( (sqrt(7) - sqrt(1))² )
= sqrt(7) - 1
原式化簡:
x + y sqrt(16 + sqrt(252)) = x sqrt(8 - sqrt(28)) + 5
x + y (3 + sqrt(7)) = x (sqrt(7) - 1) + 5
x + 3y + y sqrt(7) = x sqrt(7) - x + 5
因為 x, y 均為有理數,所以由上式得
x + 3y = -x + 5 且 y = x
解以上聯立方程式得
x + 3x = -x + 5
5x = 5
x = 1
y = 1
所以 x² + y² = 1 + 1 = 2