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Question Number 218041 by Mamadi last updated on 26/Mar/25
find (x,y,z)/such as (x+y+z=1 (x^2 +y^2 +z^2 =9 ((1/x)+(1/y)+(1/z)=1
find(x,y,z)/suchas(x+y+z=1(x2+y2+z2=9(1x+1y+1z=1
Answered by vnm last updated on 27/Mar/25
x^2 +y^2 +z^2 +2xy+2xz+2yz=1 xy+xz+yz=((1−9)/2)=−4 ((xy+xz+yz)/(xyz))=1 xyz=−4 (t−x)(t−y)(t−z)= t^3 −(x+y+z)t^2 +(xy+xz+yz)t−xyz x,y,z are the roots of t^3 −t^2 −4t+4=0 t^2 (t−1)−4(t−1)=0 (t−2)(t+2)(t−1)=0 {1, 2, −2} and all other permutations of these three numbers.
x2+y2+z2+2xy+2xz+2yz=1xy+xz+yz=1−92=−4xy+xz+yzxyz=1xyz=−4(t−x)(t−y)(t−z)=t3−(x+y+z)t2+(xy+xz+yz)t−xyzx,y,zaretherootsoft3−t2−4t+4=0t2(t−1)−4(t−1)=0(t−2)(t+2)(t−1)=0{1,2,−2}andallotherpermutationsofthesethreenumbers.

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