SOLUTION: If (a+b)²+(b+c)²+(c+d)²=4(ab+bc+cd), prove that a=b=c=d. Thanks!! :)

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


   

Question 855358: If (a+b)²+(b+c)²+(c+d)²=4(ab+bc+cd), prove that a=b=c=d.
Thanks!! :)
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
(a+b)²+(b+c)²+(c+d)²=4(ab+bc+cd), 

a²+2ab+b²+b²+2bc+c²+c²+2cd+d² = 4ab+4bc+4cd

Get 0 on the right by subtracting it from both sides:

a²-2ab+b²+b²-2bc+c²+c²-2cd+d² = 0

(a-b)²+(b-c)²+(c-d)² = 0

The three terms on the left are non-negative.

Thus they must all be 0.

Thus a-b=0, which means a=b.
     b-c=0, which means b=c,
     c-d=0, which means c=d

So a=b=c=d

Edwin