Question

Тема:Однородное показательное уравнение

Answer 1

$8*\frac{81^{x}}{72^{x}}+9*\frac{64^{x}}{72^{2}}=17*\frac{72^{x}}{72^{x}}\\\\8*(\frac{9}{8})^{x}+9*(\frac{8}{9})^{x} -17=0$

Сделаем замену :

$(\frac{9}{8})^{x}=m,m>0\Rightarrow (\frac{8}{9})^{x}=\frac{1}{m}\\\\8m+\frac{9}{m}-17=0\\\\8m^{2} -17m+9=0\\\\D=(-17)^{2} -4*8*9=289-288=1$

$m_{1}=\frac{17-1}{16}=1\\\\m_{2}=\frac{17+1}{16}=\frac{18}{16}=\frac{9}{8}\\\\1)(\frac{9}{8})^{x}=1\\\\(\frac{9}{8})^{x}=(\frac{9}{8})^{0}\\\\x_{1}=0\\\\2)(\frac{9}{8})^{x}=\frac{9}{8}\\\\x_{2} =1\\\\Otvet:\boxed{0;1}$

Сумма корней равна 1