Tìm `x^2`+`y^2`+`z^2`= xy+yz+zx

2 bình luận về “Tìm `x^2`+`y^2`+`z^2`= xy+yz+zx”

1. x^2+y^2+z^2=xy+yz+zx

   ⇔ 2x^2+2y^2+2z^2=2xy+2yz+2zx

   ⇔ (x^2-2xy+y^2)+(y^2-2yz+z^2)+(z^2-2zx+x^2)=0

   ⇔ (x-y)^2+(y-z)^2+(z-x)^2=0 

   ⇔ $\{\begin{array}{l}x-y=0\\ y-z=0\\ z-x=0\end{array}$

   ⇔ x=y=z 

   Vậy x=y=z 

    

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2. x^2 + y^2 + z^2 = xy + yz + zx

   ⇔ 2x^2 + 2y^2 + 2z^2 = 2xy + 2yz + 2zx

   ⇔ 2x^2 + 2y^2 + 2z^2 – 2xy – 2yz – 2zx = 0

   ⇔ (x^2 – 2xy + y^2) + (y^2 – 2yz + z^2) + (z^2 – 2zx + x^2) = 0

   ⇔ (x – y)^2 + (y – z)^2 + (z – x)^2 = 0

   ⇔ {(x – y = 0 ),(y – z = 0 ),(z – x = 0 ):}

   ⇔ x = y = z

   vậy x = y = z

    

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