若x,y为两个不同的实数,且满足x²=2x+1 y²=2y+1,求x^6+y^6的值
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![若x,y为两个不同的实数,且满足x²=2x+1 y²=2y+1,求x^6+y^6的值](/uploads/image/z/3994004-20-4.jpg?t=%E8%8B%A5x%2Cy%E4%B8%BA%E4%B8%A4%E4%B8%AA%E4%B8%8D%E5%90%8C%E7%9A%84%E5%AE%9E%E6%95%B0%2C%E4%B8%94%E6%BB%A1%E8%B6%B3x%26%23178%3B%3D2x%EF%BC%8B1+y%26%23178%3B%3D2y%EF%BC%8B1%2C%E6%B1%82x%5E6%2By%5E6%E7%9A%84%E5%80%BC)
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若x,y为两个不同的实数,且满足x²=2x+1 y²=2y+1,求x^6+y^6的值
若x,y为两个不同的实数,且满足x²=2x+1 y²=2y+1,求x^6+y^6的值
若x,y为两个不同的实数,且满足x²=2x+1 y²=2y+1,求x^6+y^6的值
x²-2x-1=0,y²-2y-1=0
所以可以把x、y看成关于m的方程m²-2m-1=0的两个根
根据根与系数的关系,得:x+y=2,xy=-1
所以(x+y)²=x²+y²+2xy=4,所以x²+y²=6
所以x^6+y^6=(x²+y²)(x^4-x²y²+y^4)
=(x²+y²)[(x²+y²)²-3x²y²]
=(x²+y²)[(x²+y²)²-3(xy)²]
=6×(36-3)
=198