<s>[INST] Solve this exponential equation: $3^{2x}+\left(\frac{1}{2}\right)^{-x} \cdot 3^{x+1}-2^{2x+2}=0$ I tried solving this equation $$3^{2x}+\left(\frac{1}{2}\right)^{-x} \cdot 3^{x+1}-2^{2x+2}=0$$ by taking the log of both sides, but with no results, what do I do? Sorry if this equation is very easy, I couldn't solve it...
 [/INST] we have $$(3^x)^2+2^x\cdot3^x\cdot 3-(2^x)^2\cdot2^2=0$$ divided by $(2^x)^2$ we have $$\left(\frac{3^x}{2^x}\right)^2+3\cdot \frac{3^x}{2^x}-4=0$$
setting $$u=\frac{3^x}{2^x}$$ we have a quadratic equation in $$u$$
</s>