Problem:
The sum of all real x such that (2x−4)3+(4x−2)3=(4x+2x−6)3 is
Answer Choices:
A. 3/2
B. 2
C. 5/2
D. 3
E. 7/2
Solution:
If a=2x−4 and b=4x−2 then a+b=4x+2x−6. Since (a+b)3= a3+3a2b+3ab2+b3,a3+b3 will equal (a+b)3 if and only if 3a2b+3ab2=0. Therefore,
a3+b3=(a+b)3⟺0=3ab(a+b)⟺a=0,b=0, or a+b=0.
Thus 2x−4=0 or 4x−2=0 or 4x+2x−6=(2x+3)(2x−2)=0. Since 2x+3=0 has no real roots, the sum of the real roots is 2+21+1=27.