Question

biết x+y+z=0 và xy + yz + zx = 0 , CMR x=y=z

Answer 1

Sửa đề: CMR: \(x=y=z=0\)

Ta có: \(x+y+z=0\)

\(\Rightarrow\left(x+y+z\right)^2=0\)

\(\Rightarrow x^2+y^2+z^2+2xy+2yz+2zx=0\)

\(\Rightarrow x^2+y^2+z^2+2\left(xy+yz+zx\right)=0\)

\(\Rightarrow x^2+y^2+z^2=0\) (vì \(xy+yz+zx=0\))

Ta thấy: \(\left\{{}\begin{matrix}x^2\ge0;\forall x\\y^2\ge0;\forall y\\z^2\ge0;\forall z\end{matrix}\right.\Rightarrow x^2+y^2+z^2\ge0;\forall x,y,z\)

Mà: \(x^2+y^2+z^2=0\) (cmt)

Do đó: \(\left\{{}\begin{matrix}x=0\\y=0\\z=0\end{matrix}\right.\Rightarrow x=y=z=0\) (đpcm)

$\text{#}Toru$