Log problem: Solve 25^x - 8*15^x - 9^{x+1} = 0

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bobrossu

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Problem:

Find all real numbers x such that 25x-8*15x-9x+1=0

Not sure how to start.

Steps would be helpful.
 
bobrossu said:
Problem:

Find all real numbers x such that 25x-8*15x-9x+1=0

Not sure how to start.

Steps would be helpful.
Click to expand...

\(\displaystyle 25^x - 8*15^x - 9^{x + 1} \ = \ 0\)

\(\displaystyle 25^x - 8*(5*3)^x - 9*9^x \ = \ 0\)

\(\displaystyle (5^2)^x - 8*(5^x)(3^x) - 9(3^2)^x \ = \ 0\)

\(\displaystyle (5^x)^2 - 8*(5^x)(3^x) - 9(3^x)^2 \ = \ 0\)


Pick two variables other than x, such as r and t.

Let r = \(\displaystyle \ 5^x\).

Let t = \(\displaystyle \ 3^x\).


\(\displaystyle r^2 - 8rt - 9t^2 \ = \ 0\)


Factor that trinomial, set each factor equal to zero, substitute back for what r and t equal in terms of x, and solve for any real values of x.
 
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