Question

A)2^(2x+1)+3^(2x+1)=5*6^x

Answer 1

$2^{2x+1}+ 3^{2x+1}=5*2^x*3^x$ ⇒$2* 4^{x}+3*9^x=5*2^x*3^x$, impartim ecuatia cu $9^{x}$, si se obtine ecuatia :$2*( \frac{4}{9})^x+3=5( \frac{6}{9})^x,deci:2*(( \frac{2}{3})^x)^2-5( \frac{2}{3})^x+3=0$, notam y=$(\frac{2}{3})^x$,  cu conditia y>0, si obtine ec.:$2y^2-5y+3=0.$, cu radacinile: $y_{1}=1,deci,1= (\frac{2}{3})^x,avem, x_{1}=0;si, y_{2}= \frac{3}{2}= (\frac{2}{3} )^{-1},deci. x_{2}=-1$