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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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