Question Number 217921 by Unhombre last updated on 23/Mar/25
$${solve}\:{it}: \\ $$$$\mathrm{2}+\frac{\mathrm{3}}{\mathrm{2}+\frac{\mathrm{3}}{\mathrm{2}+\frac{\mathrm{3}}{\mathrm{2}+…}}} \\ $$
Answered by Rasheed.Sindhi last updated on 24/Mar/25
$${x}=\mathrm{2}+\frac{\mathrm{3}}{\mathrm{2}+\frac{\mathrm{3}}{\mathrm{2}+\frac{\mathrm{3}}{\mathrm{2}+…}}}>\mathrm{0} \\ $$$${x}=\mathrm{2}+\frac{\mathrm{3}}{{x}} \\ $$$${x}^{\mathrm{2}} −\mathrm{2}{x}−\mathrm{3}=\mathrm{0} \\ $$$$\left({x}−\mathrm{3}\right)\left({x}+\mathrm{1}\right)=\mathrm{0} \\ $$$$\overset{\checkmark} {{x}=\mathrm{3}}\:\vee\:\overset{×} {{x}=−\mathrm{1}}\:\left({invalid}\right) \\ $$
Commented by CrispyXYZ last updated on 24/Mar/25
$$\left({x}−\mathrm{3}\right)\left({x}+\mathrm{1}\right)=\mathrm{0} \\ $$
Commented by Rasheed.Sindhi last updated on 24/Mar/25
$$\mathcal{T}{hanks}\:{sir},\:{corrected}! \\ $$