SOLUTION: Solve 24(2²ˣ) - 5(2ˣ⁺²) - 156 = 0, where x ∈ R

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Question 1210448: Solve 24(2²ˣ) - 5(2ˣ⁺²) - 156 = 0, where x ∈ R
Answer by ikleyn(53354) About Me  (Show Source):
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Solve 24(2²ˣ) - 5(2ˣ⁺²) - 156 = 0, where x ∈ R
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This given equation

    24%2A2%5E%282x%29+ -  5%2A2%5E%28x%2B2%29  - 156  =  0    (1)


is the same as

    24%5E%282x%29  -  5%2A4%2A2%5Ex  -  156  =  0,

or

    24%2A2%5E%282x%29 - +20%2A2%5Ex+ -  156  =  0.    (2)


Introduce new variable  y = 2%5Ex.


Then equation (2)  takes the form

    24y%5E2 - 20y - 156 = 0,

and we are looking for positive solutions to this equation.


Apply the quadratic formula and get the solutions to equation (2)

    y = 3  and  y = -13/6.


Only positive solution y = 3 does fit.


So, for x we have

    2%5Ex = 3,   hence,  x = log%282%2C%283%29%29 = 1.584962501  (approximately).


ANSWER.  x = log%282%2C%283%29%29 = 1.584962501  (approximately).

Solved.

This is a standard problem to solve exponential equation by introducing new variable.