Understand the problem

 .Find all real numbers x for which
$\frac{8^x+27^x}{12^x+18^x}=\frac{7}{6}$  

Source of the problem

C.M.I (Chennai mathematical institute ) U.G-2019

Topic

Real Numbers

Difficulty Level
8 out of 10

Start with hints

Do you really need a hint? Try it first!

It is of the the form of  $\frac{a^3+b^3}{a^2b+b^2a}$ . Do you observe ?  where a=\(2^x\) b=\(3^x\)

\(\frac{a^3+b^3+a^2b+b^2a}{a^2b+b^2a}\)=\(\frac{13}{6}\) \(\Rightarrow\frac{a^2+b^2}{ab}=\frac{13}{6}\) \(\frac{a}{b}+\frac{b}{a}=\frac{13}{6}\) let x=a/b then it is a quadratic equation    

\(6x^2-13x+6=0\) on solving we get x=3/2 or 2/3 now replace x by a/b

so again  putting value of a and b we get final ans is +1 and -1 

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