Question Number 92283 by jagoll last updated on 06/May/20
$$\mathrm{9}^{\mathrm{x}} +\mathrm{3}^{\mathrm{x}} \:=\:\mathrm{25}^{\mathrm{x}} −\mathrm{5}^{\mathrm{x}} \: \\ $$$$\mathrm{find}\:\frac{\mathrm{5}^{\mathrm{x}} }{\mathrm{3}^{\mathrm{x}} +\mathrm{1}}\:? \\ $$
Commented by john santu last updated on 06/May/20
$$\left(\mathrm{3}^{{x}} \right)^{\mathrm{2}} +\left(\mathrm{3}^{{x}} \right)−\left(\mathrm{5}^{{x}} \right)^{\mathrm{2}} +\left(\mathrm{5}^{{x}} \right)=\mathrm{0} \\ $$$${let}\:\mathrm{3}^{{x}} \:=\:{u}\:;\:\mathrm{5}^{{x}} ={v} \\ $$$$\left({u}−{v}\right)\left({u}+{v}\right)+\left({u}+{v}\right)=\mathrm{0} \\ $$$$\left({u}+{v}\right)\left({u}−{v}+\mathrm{1}\right)= \\ $$$$\Rightarrow\:{u}\:=\:{v}−\mathrm{1}\:,\:{v}\:=\:{u}+\mathrm{1} \\ $$$$\mathrm{5}^{{x}\:} \:=\:\mathrm{3}^{{x}} +\mathrm{1}\:\Rightarrow\:\frac{\mathrm{5}^{{x}} }{\mathrm{3}^{{x}} +\mathrm{1}}\:=\:\mathrm{1} \\ $$$$ \\ $$
Commented by jagoll last updated on 06/May/20
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Commented by $@ty@m123 last updated on 06/May/20
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