SOLUTION: Determine all real values of x for which {{{3^(x+2)+2^(x+2)+2^x=2^(x+5)+3^x}}}

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Question 1114515: Determine all real values of x for which 3%5E%28x%2B2%29%2B2%5E%28x%2B2%29%2B2%5Ex=2%5E%28x%2B5%29%2B3%5Ex
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
x = 3 by graphing
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3%5E%28x%2B2%29%2B2%5E%28x%2B2%29%2B2%5Ex=2%5E%28x%2B5%29%2B3%5Ex
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3%5E5+%2B+2%5E5+%2B+2%5E3=+2%5E8+%2B+3%5E3
243 + 32 + 8 = 256 + 27
283 = 283

Answer by ikleyn(53763) About Me  (Show Source):
You can put this solution on YOUR website!
.
3%5E%28x%2B2%29%2B2%5E%28x%2B2%29%2B2%5Ex = 2%5E%28x%2B5%29%2B3%5Ex  ====>


3%5E%28x%2B2%29+-+3%5Ex = 2%5E%28x%2B5%29+-+2%5E%28x%2B2%29+-+2%5Ex   ====>


3%5Ex%2A%283%5E2-1%29 = 2%5EX%2A%282%5E5-2%5E2-1%29  ====>


3%5Ex%2A8 = 2%5Ex%2A27.   (*)


At this point, you can guess at least one root  x = 3.


Next, transforming (*) into


%283%2F2%29%5Ex = 27%2F8,


you can see that this root is unique: there are NO other roots.