Question

Sa se rezolve in R ecuația
$3 \times 4 ^{x} + 2 \times {9}^{x} = 5 \times {6}^{x}$

Answer 1

$3\cdot 4^x+2\cdot 9^x = 5\cdot 6^x\\3\cdot 2^{2x}-5\cdot 2^x\cdot 3^x +2\cdot 3^{2x}= 0| :3^{2x} \neq 0\\3\left( \dfrac{2}{3}\right)^{2x} -5\cdot \left( \dfrac{2}{3}\right)^x +2=0\\\left(\dfrac{2}{3}\right)^x \stackrel{not}{=} t, t>0\\3t^2-5t+2=0\\\Delta= 25-24=1\Rightarrow \sqrt {\Delta}=1\\t_1=\dfrac{5-1}{6}= \dfrac{2}{3} \\t_2=\dfrac{5+1}{6}=  1\\\left(\dfrac{2}{3}\right)^x = \dfrac{2}{3}\Rightarrow x=1\\\left(\dfrac{2}{3}\right)^x = 1\Rightarrow x=0\\S:x\in \{0,1\}$