已知2/x=3/y=4/z,求4x^2+2yz+z^2/x+y+z*x-z-y/8x^2+4yz+2z^2的值
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已知2/x=3/y=4/z,求4x^2+2yz+z^2/x+y+z*x-z-y/8x^2+4yz+2z^2的值
![已知2/x=3/y=4/z,求4x^2+2yz+z^2/x+y+z*x-z-y/8x^2+4yz+2z^2的值](/uploads/image/z/18162913-49-3.jpg?t=%E5%B7%B2%E7%9F%A52%2Fx%3D3%2Fy%3D4%2Fz%2C%E6%B1%824x%5E2%2B2yz%2Bz%5E2%2Fx%2By%2Bz%2Ax-z-y%2F8x%5E2%2B4yz%2B2z%5E2%E7%9A%84%E5%80%BC)
两种方法:
1、一般法
设2/x=3/y=4/z=1/k,则x=2k,y=3k,z=4k,于是
4x^2+2yz+z^2/x+y+z*x-z-y/8x^2+4yz+2z^2
=[4(2k)^2+2*3k*4k+(4k)^2]/(2k+3k+4k) * (2k-4k-3k)/[8(2k)^2+4*3k*4k+2(4k)^2]
=(56k^2)/(9k) * (-5k)/[(112k^2)
=56k/9*(-5/112k)
=-5/18
2、特殊值法:令x=2,y=3,z=4代入上式可求(略).
方法3:先化简,再求值
4x^2+2yz+z^2/x+y+z*x-z-y/8x^2+4yz+2z^2
=4x^2+2yz+z^2/x+y+z*x-z-y/2(4x^2+2yz+z^2)
=(x-z-y)/2(x+y+z)
=-5/18
1、一般法
设2/x=3/y=4/z=1/k,则x=2k,y=3k,z=4k,于是
4x^2+2yz+z^2/x+y+z*x-z-y/8x^2+4yz+2z^2
=[4(2k)^2+2*3k*4k+(4k)^2]/(2k+3k+4k) * (2k-4k-3k)/[8(2k)^2+4*3k*4k+2(4k)^2]
=(56k^2)/(9k) * (-5k)/[(112k^2)
=56k/9*(-5/112k)
=-5/18
2、特殊值法:令x=2,y=3,z=4代入上式可求(略).
方法3:先化简,再求值
4x^2+2yz+z^2/x+y+z*x-z-y/8x^2+4yz+2z^2
=4x^2+2yz+z^2/x+y+z*x-z-y/2(4x^2+2yz+z^2)
=(x-z-y)/2(x+y+z)
=-5/18
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