Any ideas on how to accomplish completing this square: $\displaystyle x^2+y^2+z^2-xy-xz-yz=0$. I've tried a few things, but I keep getting additional terms. Thanks.
Hello, brucewayne!
Any ideas on how to complete this square? . $\displaystyle x^2+y^2+z^2-xy-yz-xz\:=\:0$
$\displaystyle \text{Multiply by 2: }\:2x^2 + 2y^2 + 2z^2 - 2xy - 2yz - 2xz \:=\:0$
. . $\displaystyle (x^2 - 2xy + y^2) + (y^2 - 2yz + z^2) + (x^2 - 2xz + z^2) \:=\:0$
. . . . . . . . $\displaystyle (x-y)^2 + (y-z)^2 + (x-z)^2 \:=\:0$
$\displaystyle \text{We have: \:the sum of three squares equals zero.}$
$\displaystyle \text{This is true when: }\:x \,=\,y\,=\,z$