Question

已知14(XX+YY+ZZ)=(X+2Y+3Z)^2,求X:Y:Z

Answer 1

X:Y:Z=1:2:3

因为：14(XX+YY+ZZ)=(X+2Y+3Z)^2

14(XX+YY+ZZ)-(X+2Y+3Z)^2=0

13X^2+10Y^2+5Z^2-4XY-6XZ-12YZ=0

(4X^2-4XY+Y^2)+(9X^2-6XZ+Z^2)+(9Y^2-12YZ+4Z^2)=0

(2X-Y)^2+(3X-Z)^2+(3Y-2Z)^2=0

所以2X-Y=0,3X-Z=0,3Y-2Z=0

X:Y=1:2,X:Z=1:3,Y:Z=2:3

所以X:Y:Z=1:2:3