a. Aratati ca x²+y²+z²-(xy+xz+yz)=1supra 2[(x-y)²+(x-z)²+(y-z)²] oricare ar fi x,y,z∈R b.aratati ca daca x²+y²+x²=xy+xz+yz,atunci. x=y=z c.determinati multimea{(a,b)}∈R² Ι a²+b²+4=ab+2a+2b}
a)[tex]\frac{1}{2}[(x-y)^2+(x-z)^2+(y-z)^2]=\\
=\frac{1}{2}(x^2-2xy+y^2+x^2-2xz+z^2+y^2-2yz+z^2)=\\
=\frac{1}{2}[2x^2+2y^2+2z^2-2(xy+xz+yz)]=\\
=x^2+y^2+z^2-(xy+xz+yz)[/tex] b)Folosind relatia demonstrata de la punctul a): [tex]\frac{1}{2}[(x-y)^2+(x-z)^+(y-z)^2]=0\\
x-y=0=>x=y\\
x-z=0=>x=z\\
y-z=0=>y=z\\
x=y=z
[/tex] c) Folosind cele demonstrate la punctul b) din: [tex]a^2+b^2+2^2=ab+2a+2b=>a=b=2\\
M=\{(2;2)\}[/tex]
