What is the solution of the equation 2^(1+x)-2^(3-x)=15 ?

We'll re-write the terms of the equation, knowing that the
exponents are added when we multiply two powers that share the same base and the
exponents are subtracted when we divide two powers that have matching
bases:

2*`2^(x)` - `2^(3)` *`2^(-x)` =
15

We'll use the negative power
property:

`2^(-x)` =
1/`2^(x)`

We'll re-write the
equation:

2*`2^(x)` - 8/`2^(x)` =
15

We'll multiply both sides by `2^(x)`
:

2*`2^(2x)` - 8 - 15*`2^(x)` =
0

We'll replace `2^(x)` by
t:

`t^(2)` - 15t - 8 =
0

We'll apply quadratic
formula:

`t_(1,2)` = (15`+-` `sqrt(225 + 32)`
)/2

`t_(1,2)` = (15`+-` `sqrt(257)`
)/2

t1 = 15.51

t2 =
-0.51

But `2^(x)` = t => `2^(x)` = 15.51 =>
x*ln 2 = ln 15.51

x = ln 15.51/ln
2

x = 2.74/0.69

x =
3.97

`2^(x)` = -0.51 impossible because `2^(x)` > 0
for any real value of x.

Therefore, the
solution of the equation is x = 3.97.