Solve the following equation(without utilising calculus): $$ 2005^x-2004^x=1+ 3 \cdot (2004^{\frac{x}{3}} +2004^{\frac{2x}{3}}) $$
I've tried switching everything to the other side of the inequality except $ 2005^x$ to then divide by it and have observed the function resulted from it is always positive but I haven't been able to progress further; I should find the intersection of said function $ f(x)=\frac{1}{2005^x} + (\frac{2004^{1/3}}{2005})^x +(\frac{2004^{ 2/3}}{2005})^x+ (\frac{2004}{2005})^x $ with $1$ to solve the equation.(There probably are better methods than this which I'm unaware of).