(a^4 + b^4) x^2 +4abcd x +(c^4+d^4) =0 prove that the roots of the equation cannot be different if real
medha , 8 Years ago
Grade 12th pass
Algebra> (a^4 + b^4) x^2 +4abcd x +(c^4+d^4) =0 pr...
2 AnswersNicho priyatham
Last Activity: 8 Years ago
Given Roots are real so discriminent >0
(4abcd)2-4(a2+b2)(c2+d2)>0
so 4(abcd)2 >(a2+b2)(c2+d2)
((a/b)2+(b/a)2)((c/d)2+(d/c)2)4 .........(1)
from A.M>G.M we get
so we get (a/b)2+(b/a)2)>2 and ((c/d)2+(d/c)2>2
so ((a/b)2+(b/a)2)((c/d)2+(d/c)2) >4 …......(2)
from equation (1) and (2)
we get ((a/b)2+(b/a)2)((c/d)2+(d/c)2) =4
so only possible if a=b and c=d
which gives discriminent =0 hence equal if real
plz APPROVE if usefull
incase of any doubts drop them down in Answer box
Vikas TU
Last Activity: 8 Years ago
Calculate the Discriminant ,
D = (4abcd)^2 – 4*(a^4 + b^4)*(c^4+d^4)
Take a=b=c=d = 1
D = zero for all abcd being real.
Take a = b =c = d= ½
D
U can take any real no. disitinct for abcd
D
Hence D is never positive.
Hence roots never be positive.
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