Solve the equation 2^x + 4^(x+1)/2 = 12

To solve

2^x+4^(x+1)/2 =
12.

Rewrite as:

2^x+4*4^x /2 =
12

2^x+2*(2^x)^2 = 12 , as 4^x = (2^2)^x =
(2^x)^2

y +2y^2-12 = 0, where y =
2^x.

2y^2 +y-12 = 0

 y =
{-1+sqrt(1+4*2*12)}/4 

y  =
{-1+sqrt97}/4

 y =
{-1-sqrt97}/4

Therefore  2  ^x =
(-1+sqrt97)/4

2log2 = {log(-1+sqrt97/4} / log
2

x = log {(-1+sqrt97)/4 }/
log2.

But if we take the middle term, 4^(x+1)/2
= 4^((x+1)/2):

 4 = 2^2.  So 4^((x+1)/2) = 2^2((x+1)/2) =
2^(x+1).

Now the equation could be rewritten
as:

2^x+2^(x+1) =
12

(2^x)(1+2)  = 12

(2^x)(3)
=12

2^x = 12/3 = 4 = 2^2

2^x =
2^2

x = 2