Solve the equation 2^x - 14*2^-x=-5?show the steps

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to solve the equation: 2^x - 14*2^-x = -5

2^x - 14*2^-x = -5

Let 2^x = y

=> y - 14/y + 5 = 0

=> y^2 + 5y - 14 = 0

=> y^2 + 7y - 2y - 14 = 0

=> y(y + 7) - 2(y + 7) = 0

=> (y - 2)(y + 7) = 0

=> y = 2 and y = -7

as y = 2^x

2^x = 2

=> x = 1

2^x = -7 does not provide a solution for x as 2 and 2^x cannot be negative.

The required solution of the equation is x = 1.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

First, we'll re-write the equation, using the negative power property of exponential functions.

`2^x - 14/(2^x)=-5`

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

`2^2x - 14 = -5*2^x`

We'll move all terms to one side:

`2^2x + 5*2^x - 14 = 0`

We'll treat the equation above as a quadratic equation and we'll solve it using quadratic formula, `2^x ` being the unknown:

`2^x = (-5+sqrt(25 + 56))/2`

`` `2^x = (-5+sqrt81)/2`

`2^x = (-5+9)/2`

`2^x = 2`  => x = 1

`2^x = (-5-9)/2`

`2^x`  = -7 impossible since 2^x > 0 for any real value of x.

Therefore, the equation will have only one solution: x = 1.

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