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Mugurelantonesa
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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}

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Răspuns :

Matepentrutoti
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]