Question Number 228440 by fantastic2 last updated on 14/Apr/26
$$\mathrm{3}^{{x}} +\mathrm{9}^{{x}} +\mathrm{27}^{{x}} =\mathrm{14} \\ $$
Answered by Lara2440 last updated on 15/Apr/26
$$\mathrm{3}^{{x}} \:\overset{\mathrm{Substitution}} {=}{u}\:,\:{u}>\mathrm{0} \\ $$$${u}^{\mathrm{3}} +{u}^{\mathrm{2}} +{u}=\mathrm{14} \\ $$$${f}\left({u}\right)={u}^{\mathrm{3}} +{u}^{\mathrm{2}} +{u}−\mathrm{14} \\ $$$${f}\left(\mathrm{2}\right)=\mathrm{0} \\ $$$$\therefore\:{f}\left({u}\right)=\left({u}−\mathrm{2}\right)\left(\mathrm{bla}\:\mathrm{bla}\right) \\ $$$$\mathrm{2}\lfloor \\ $$
Commented by fantastic2 last updated on 14/Apr/26
$${great} \\ $$
Commented by Lara2440 last updated on 15/Apr/26
$${THX}\heartsuit \\ $$
Commented by Lara2440 last updated on 15/Apr/26
$$\mathrm{I}\:\mathrm{fixed}\:\mathrm{a}\:\mathrm{few}\:\mathrm{minor}\:\mathrm{logical}\:\mathrm{inconsistencies} \\ $$$$\mathrm{but}\:\mathrm{I}'\mathrm{m}\:\mathrm{not}\:\mathrm{enirely}\:\mathrm{sure}\:\mathrm{if}\:\mathrm{the}\:\mathrm{over}\:\mathrm{all}\:\mathrm{solution}\:\mathrm{is}\:\mathrm{correct}.\: \\ $$$$\mathrm{The}\:\mathrm{equation}\:\mathrm{holds}\:\mathrm{due}\:\mathrm{to}\:\mathrm{the}\:\mathrm{laws}\:\mathrm{of}\:\mathrm{exponents}\:\mathrm{and}\: \\ $$$$\mathrm{one}-\mathrm{to}-\mathrm{one}\:\mathrm{correspondence} \\ $$$$\mathrm{but}\:\mathrm{I}\:\mathrm{omitted}\:\mathrm{those}\:\mathrm{details}\:\mathrm{to}\:\mathrm{keep}\:\mathrm{it}\:\mathrm{clean}. \\ $$$$\mathrm{It}\:\mathrm{was}\:\mathrm{an}\:\mathrm{interesting}\:\mathrm{problem} \\ $$$$\mathrm{Are}\:\mathrm{there}\:\mathrm{anymore}\:\mathrm{challenging}\:\mathrm{ones}???? \\ $$
Commented by fantastic2 last updated on 15/Apr/26
$${i}\:{will}\:{post}\:{if}\:{i}\:{get}\:{one} \\ $$