已知x-y=5,y-z=4,求xx+yy+zz-xy-yz-xz的值

x-z=(x-y)+(y-z)=5+4=9

(x-y)²=x²-2xy+y²=25
(y-z)²=y²-2yz+z²=16
(x-z)²=x²-2xz+z²=81

上面的三个式子相加得
x²-2xy+y²+y²-2yz+z²+x²-2xz+z²=25+16+81
2(xx+yy+zz-xy-yz-xz)=122
xx+yy+zz-xy-yz-xz=61

x-z=(x-y)+(y-z)=

xx+yy+zz-xy-yz-xz
=0.5*[(x-y)^2+(y-z)^2+(z-x)^2]
=0.5*(5^2+4^2+9^2)
=0.5*122
=61
x-y+y-z=x-z=9

x-y=5①
y-z=4② ①+②，得
x-z=9③
①*②,得xy-yy-xz+yz=20④
①*③,得xx-xy-xz+yz=45⑤
②*③,得xy-yz-xz+zz=36⑥
⑤+⑥-④=xx+yy+zz-xy-yz-xz=61