[tex]\it 3\cdot4^{x} +2\cdot9^{x} = 5\cdot6^x \Leftrightarrow 3\cdot2^{2x} +2\cdot3^{2x}=5\cdot2^x\cdot3^x \Leftrightarrow
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\Leftrightarrow 3\cdot2^{2x} +2\cdot3^{2x}=3\cdot2^x\cdot3^x +2\cdot2^x\cdot3^x\Leftrightarrow
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\Leftrightarrow 3\cdot2^{2x} +2\cdot3^{2x}-3\cdot2^x\cdot3^x -2\cdot2^x\cdot3^x = 0\Leftrightarrow [/tex]
[tex]\it 3\cdot2^x(2^x-3^x) -2\cdot3^x(2^x-3^x) =0 \Leftrightarrow
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\Leftrightarrow (2^x-3^x)(3\cdot2^x-2\cdot3^x)=0 \Leftrightarrow \begin{cases} \it 2^x-3^x=0 \ \ \ \ \ (1)
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\it3\cdot2^x-2\cdot3^x = 0 \ \ \ (2)\end{cases}[/tex]
[tex]\it (1) \Rightarrow 2^x=3^x \Rightarrow x = 0
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(2) \Rightarrow3\cdot2^x=2\cdot3^x \Rightarrow \dfrac{2^x}{3^x}=\dfrac{2}{3} \Rightarrow \left(\dfrac{2}{3}\right)^x=\dfrac{2}{3} \Rightarrow x = 1[/tex]
Mulțimea soluțiilor ecuației este:
S = {0, 1}
