已知实数x,y,z满足x+y+z=5,xy+yz+zx=3.求z的取值范围
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已知实数x,y,z满足x+y+z=5,xy+yz+zx=3.求z的取值范围
![已知实数x,y,z满足x+y+z=5,xy+yz+zx=3.求z的取值范围](/uploads/image/z/5630388-60-8.jpg?t=%E5%B7%B2%E7%9F%A5%E5%AE%9E%E6%95%B0x%2Cy%2Cz%E6%BB%A1%E8%B6%B3x%2By%2Bz%3D5%2Cxy%2Byz%2Bzx%3D3.%E6%B1%82z%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
x+y=5-z
xy+yz+zx=3 xy+(x+y)z=3 xy=3-(x+y)z=3-(5-z)z=z^2-5z+3
构建 一个以x和y为根的一元二次方程
m^2-(5-z)m+z^2-5z+3=0
Δ=(5-z)^2-4(z^2-5z+3)=-3z^2+10z+13≥0
所以 -1≤z≤13/3
z的取值范围是-1≤z≤13/3
xy+yz+zx=3 xy+(x+y)z=3 xy=3-(x+y)z=3-(5-z)z=z^2-5z+3
构建 一个以x和y为根的一元二次方程
m^2-(5-z)m+z^2-5z+3=0
Δ=(5-z)^2-4(z^2-5z+3)=-3z^2+10z+13≥0
所以 -1≤z≤13/3
z的取值范围是-1≤z≤13/3
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