Solve: \(3^{2x+4} + 1 = 2\cdot3^{x+2}\)
Solution
\[ 3^{2x+4} + 1 = 2\cdot3^{x+2} \]
\[ \text{Let } t = 3^{x+2} \]
\[ \Rightarrow t^2 + 1 = 2t \]
\[ \Rightarrow t^2 – 2t + 1 = 0 \]
\[ \Rightarrow (t – 1)^2 = 0 \]
\[ \Rightarrow t = 1 \]
\[ \Rightarrow 3^{x+2} = 1 = 3^0 \]
\[ \Rightarrow x + 2 = 0 \]
\[ \Rightarrow x = -2 \]
Final Answer:
\[ \boxed{x = -2} \]