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Find all possitive solutions of the equation:
(1 +5√3)^x + (1 +4√3)^x = 2 +9√3
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Solve?
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by keentoknow » Sat May 05, 2007 1:17 pm
Well obviously x=1 works
if x>1 (1 +5√3)^x>1 +5√3
and (1 +4√3)^x>1 +4√3
add this to obtain (1 +5√3)^x + (1 +4√3)^x > 2 +9√3 for x>1.
if x<1 (1 +5√3)^x + (1 +4√3)^x < 2+9√3 for x<1 (the same reasoning).
So x=1 is the only solution!
if x>1 (1 +5√3)^x>1 +5√3
and (1 +4√3)^x>1 +4√3
add this to obtain (1 +5√3)^x + (1 +4√3)^x > 2 +9√3 for x>1.
if x<1 (1 +5√3)^x + (1 +4√3)^x < 2+9√3 for x<1 (the same reasoning).
So x=1 is the only solution!
- keentoknow
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