Question

Sa se rezolve ecuatia
3×4^x+2×9^x=5×6^x

Answer 1

[tex]\it 3\cdot4^{x} +2\cdot9^{x} = 5\cdot6^x \Leftrightarrow 3\cdot2^{2x} +2\cdot3^{2x}=5\cdot2^x\cdot3^x \Leftrightarrow \\\;\\ \Leftrightarrow 3\cdot2^{2x} +2\cdot3^{2x}=3\cdot2^x\cdot3^x +2\cdot2^x\cdot3^x\Leftrightarrow \\\;\\ \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 \\\;\\ \Leftrightarrow (2^x-3^x)(3\cdot2^x-2\cdot3^x)=0 \Leftrightarrow \begin{cases} \it 2^x-3^x=0 \ \ \ \ \ (1) \\ \\ \it3\cdot2^x-2\cdot3^x = 0 \ \ \ (2)\end{cases}[/tex]

[tex]\it (1) \Rightarrow 2^x=3^x \Rightarrow x = 0 \\ \\ (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}