Problem: The equality (x+m)2−(x+n)2=(m−n)2, where m and n are unequal nonzero constants, is satisfied by x=am+bn where:
Answer Choices:
A. a=0, b​ has a unique non-zero value
B. a=0,b​ has two non-zero values
C. b=0, a​ has a unique non-zero value
D. b=0, a​ has two non-zero values
E. a​ and b​ each have a unique non-zero value.
Solution:
(x+m)2−(x+n)2=(m−n)2
Factor the left side of the equation as the difference of two squares.
(x+m−x−n)(x+m+x+n)=(m−n)2
(m−n)(2x+m+n)=(m−n)2
Since mî€ =n;2x+m+n=m−n,x=−n, so that (A) is the correct choice.