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Find all solutions to 4^x - 2^x = 56 + 11*2^x + 2^(x - 1).

 May 1, 2022
 #1
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Only 4

 May 1, 2022
 #2
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\(4^x - 2^x = 56 + 11\cdot 2^x + 2^{x - 1}\)

 

If we let \(t = 2^x\) we have 

\(t^2 - t = 56 + 11t + \dfrac12 t\\ 2t^2 - 25t - 112 = 0\\ (t - 16)(2t + 7) = 0\)

 

It is not possible that 2^x = -7/2, therefore 2^x = 16, which corresponds to x = 4.

 May 2, 2022

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