Question

Cum se rezolva această problemă?​

Answer 1

$3\cdot 4^x+2\cdot 9^x=5\cdot 6^x \\ \\3\cdot 2^{2x}+2\cdot 3^{2x} = 5\cdot 2^x\cdot 3^x\\ \\ 2^x = a>0,\quad 3^x = b>0\\ \\ 3a^2+2b^2 = 5ab\Big|:(ab) \\ \\ \dfrac{3a}{b}+\dfrac{2b}{a} = 5 \\ \\ \text{Notez: }\quad\dfrac{a}{b} = t>0 \\ \\ 3t+\dfrac{2}{t} = 5 \\ \\ 3t^2-5t+2 = 0$

$\Delta = 25 - 24 = 1 \Rightarrow t_{1,2} = \dfrac{5\pm 1}{6} \Rightarrow \left|\begin{array}{lcl}t =\dfrac{2}{3} \\ \\ t =1\end{array}\right. \Rightarrow \\ \\ \Rightarrow \left|\begin{array}{lcl}\dfrac{2^x}{3^x} =\dfrac{2}{3} \\ \\ \dfrac{2^x}{3^x} =1\end{array}\right. \Rightarrow\left|\begin{array}{lcl} x = 1 \\ \\ x = 0\end{array}\right. \Rightarrow \boxed{x\in \big\{0,1\big\}}$