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If a ,b,c are positive real numbers such that the equations a(x^2) + bx + c =0 and b(x^2) + cx + a = 0 , have a common root, then 1)a + bw + c(w^2) = 0 2)a + b(w^2)+ cw = 0 3)a^3 + b^3 + c^3 = 3abc 4)all of the above


If a ,b,c are positive real numbers such that the equations a(x^2) + bx + c =0 and b(x^2) + cx + a =  0 , have a common root, then 
1)a + bw + c(w^2) = 0
2)a + b(w^2)+ cw = 0
3)a^3 + b^3 + c^3 = 3abc
4)all of the above
 

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1 Answers

Aditya Gupta
2081 Points
5 years ago
let the common root be w.
then a(w^2) + bw + c =0 
and b(w^2) + cw + a =  0 
subtracting,
(a-b)(w^2) +(b-c)w + c-a =  0 
so w=1 or (c-a)/(a-b)
if w=1, then a+b+c=0, but a, b and c are positive so this cant be true.
if w=(c-a)/(a-b), then on putting this value in the original equation, and simplifying we get
a^2+b^2+c^2-ab-bc-ca=0
or [a + bw + c(w^2)][a + b(w^2)+ cw ]=0
so, either  a + b(w^2)+ cw =0, or a + bw + c(w^2)=0.and 
a^3 + b^3 + c^3 –3abc= (a+b+c)(a + b(w^2)+ cw)( a + bw + c(w^2))=0
note that if a + bw + c(w^2) =0, that automatically implies )a + b(w^2)+ cw =0 [taking conjugate]
so, so, option (4) is correct.

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