Question

Find all real numbers x satisfying the equation:

2^x + 3^x - 4^x + 6^x - 9^x = 1.

Please help me. I really don't know what to do.

Answer 1

Hmm. It's clear that x=0 works, but is there anything else?

(2^x+1)(3^x+1) = 2^x + 3^x + 6^x + 1
so,
(2^x+1)(3^x+1) +1-4^x + 1-9^x = 4

(1+2^x)(1+3^x) + (1+2^x)(1-2^x) + (1+3^x)(1-3^x) = 4

(1+2^x)(1+3^x+1-2^x) + (1+3^x)(1-3^x) = 4

I don't see anything dropping out. If we can show that 1 is the maximum value of the function, then there is probably no other solution. Can't see right off how to show that, either. There's probably some clever trick involved.

Answer 2

Thanks, I already know that 0 is the answer but I don't know how to arrive to that. Huhu, I don't know the solution. Please help me, thanks.

Answer 3

A basketball player scored 26 times during a game. He scored a total of 45 points, 2 for each field goal and 1 for each free throw. How many 2 point shots did he make and how many free throws?