7.214. $9^{x}+6^{x}=2^{2 x+1}$.

Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.

7.214. $9^{x}+6^{x}=2^{2 x+1}$.
Solution.

Rewrite the equation as $3^{2 x}+2^{x} \cdot 3^{x}-2 \cdot 2^{2 x}=0$ and divide it by $2^{2 x} \neq 0$. Then $\left(\frac{3}{2}\right)^{2 x}+\left(\frac{3}{2}\right)^{x}-2=0 \Rightarrow\left(\left(\frac{3}{2}\right)^{x}\right)=-2$ (no solutions) or $\left(\left(\frac{3}{2}\right)^{x}\right)_{2}=1 \Rightarrow x=0$.

Answer: 0.