The sum of all real x such that &(2^x-4)^3+(4^x-2)^3 & =(4^x+2^x-6)^3 is

1991 AMC 12 Problem 20

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20.

The sum of all real

x

such that

\begin{aligned} &(2^x-4)^3+(4^x-2)^3\\ &\qquad=(4^x+2^x-6)^3 \end{aligned}

\frac32

2

\frac52

3

\frac72

Set

a=2^x-4

and

b=4^x-2

Factor

a^3+b^3-(a+b)^3

-3ab(a+b)

Let

a=2^x-4

and

b=4^x-2.

The right side is

(a+b)^3,

a^3+b^3=(a+b)^3

is equivalent to

ab(a+b)=0.

The cases give

2^x=4,

4^x=2,

4^x+2^x=6,

whose real solutions are

x=2,\frac12,

and

1,

respectively. Their sum is

2+\frac12+1=\frac72.

Thus the correct answer is E.