SOLUTION: If (a+b)^2 + (b+c)^2 + (c+d)^2 = 4(ab+bc+cd), prove a=b=c=d

Algebra ->  Equations -> SOLUTION: If (a+b)^2 + (b+c)^2 + (c+d)^2 = 4(ab+bc+cd), prove a=b=c=d      Log On


   

Question 742010: If (a+b)^2 + (b+c)^2 + (c+d)^2 = 4(ab+bc+cd), prove a=b=c=d
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
%28a%2Bb%29%5E2+%2B+%28b%2Bc%29%5E2+%2B+%28c%2Bd%29%5E2+=+4%28ab%2Bbc%2Bcd%29

a%5E2%2B2ab%2Bb%5E2%2Bb%5E2%2B2bc%2Bc%5E2%2Bc%5E2%2B2cd%2Bd%5E2=4ab%2B4bc%2B4cd

a%5E2%2B2ab%2Bb%5E2%2Bb%5E2%2B2bc%2Bc%5E2%2Bc%5E2%2B2cd%2Bd%5E2-4ab-4bc-4cd=0

a%5E2-2ab%2Bb%5E2%2Bb%5E2-2bc%2Bc%5E2%2Bc%5E2-2cd%2Bd%5E2=0

%28a-b%29%5E2%2B%28b-c%29%5E2%2B%28c-d%29%5E2=0
so (a-b)^2=0
a-b=0
a=b
similarly b=c
c=d
a=b=c=d