SOLUTION: Find all values of K such that equation 2^x-k(2^-x)=1 has exact one real solution

Question 1011381: Find all values of K such that equation 2^x-k(2^-x)=1 has exact one real solution
Answer by ikleyn(52788) (Show Source): You can put this solution on YOUR website!
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Find all values of k such that equation 2^x-k(2^-x)=1 has exactly one real solution.
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 = .      (1)

Introduce new variable y = . Then the equation takes the form

y - k* = .

Multiply both sides by y. You will get

 = ,    or

 = .         (2)

The condition that this equation has only one root is vanishing the discriminant, i.e. 

d = 0,

where d =  =  = .

It means that 1 + 4k = 0,   or   k = .

Then the equation (2) become 

 = .         (3)

The discriminant of this equation is zero due to selection of k. (You can check it).

The unique root of this equation is y = .

Thus the equation (1) has the unique root if and only if k = . 

This root is x = -1.

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