Question

решите уравнение
4^x+1-13*6^x+9^x+1=0

Answer 1

$4^{x+1}-13cdot 6^x+9^{x+1}=0\\4cdot 4^x-13cdot 6^x+9cdot 9^x\\4cdot 2^{2x}-13cdot 2^xcdot 3^x+9cdot 3^{2x}=0\\\text{Delim vse na } 2^xcdot 3^x\\\4cdot dfrac{2^x}{3^x} -13+9cdot dfrac{3^x}{2^x} =0\\\t:= bigg(dfrac{2}{3} bigg)^x geq 0\\\4t+ dfrac{9}{t} -13=0$

$4t+ dfrac{9}{t} -13=0\\4t^2-13t+9=0\\D=25\\t_1= dfrac{13+5}{8} = dfrac{9}{4} ;quad t_2=1$
 
$bigg(dfrac{2}{3} bigg)^x= dfrac{9}{4}\\bigg(dfrac{2}{3} bigg)^x= bigg(dfrac{2}{3}bigg)^{-2}\\x=-2$

$bigg(dfrac{2}{3} bigg)^x=1\\x=0$