Задача 55507 помогите решить показательное уравнение...

5f9977d26feaba08a85df21e

11.11.2020 19:08:07

помогите решить показательное уравнение

5f3ea7e3faf909182968ddd9

11.11.2020 19:38:52

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[m]27^{\frac{2}{3}}\cdot 27^{x}+5\cdot 8^{x}\cdot 8^{\frac{1}{3}}=19\cdot 3^{x}\cdot 4^{x}[/m]

[m]((3^3){\frac{1}{3}})^2\cdot (3^{3})^{x}+5\cdot (2^3)^{x}\cdot (2^3)^{\frac{1}{3}}=19\cdot 3^{x}\cdot (2^2)^{x}[/m]

[m]9\cdot (3^{x})^{3}+5\cdot (2^{x})^{3}\cdot 2=19\cdot 3^{x}\cdot (2^{x})^{2}[/m]

Делим на [m](2^{x})^{3}[/m]

[m]9\cdot ((\frac{3}{2})^{x})^3- 19\cdot(\frac{3}{2})^{x}+10=0[/m]

[m]9\cdot ((\frac{3}{2})^{x})^3- 9\cdot(\frac{3}{2})^{x}-10\cdot(\frac{3}{2})^{x}+10=0[/m]


[m]9\cdot((\frac{3}{2})^{x})^3-9\cdot ( \frac{3}{2})^{x}-10\cdot((\frac{3}{2})^{x}-1)=0[/m]

[m]9\cdot (\frac{3}{2})^{x})\cdot((\frac{3}{2})^{x})^2-1)-10\cdot((\frac{3}{2})^{x}-1)=0[/m]

[m]9\cdot (\frac{3}{2})^{x})\cdot((\frac{3}{2})^{x}-1)\cdot ((\frac{3}{2})^{x})+1)-10\cdot((\frac{3}{2})^{x}-1)=0[/m]


[m]((\frac{3}{2})^{x}-1)\cdot (9\cdot ((\frac{3}{2})^{x})^2+9\cdot (\frac{3}{2})^{x} -10)=0[/m]


[m](\frac{3}{2})^{x}-1=0 [/m] или [m] 9\cdot ((\frac{3}{2})^{x})^2+9\cdot (\frac{3}{2})^{x} -10=0[/m]; D=81+4*9*10=441=21^2

[m](\frac{3}{2})^{x}=1 [/m] или [m](\frac{3}{2})^{x}=\frac{2}{3} [/m]или [m](\frac{3}{2})^{x}=-\frac{5}{3} [/m] ( нет корней)

[m] x=0 [/m] или [m] x=-1 [/m]