Question

Rezolvati in R ecuatia
$( \frac{2}{3} )^{x+1} +( \frac{2}{3} )^{1-x}= \frac{13}{9}$

Answer 1

Ecuatia se poate scrie: $\frac{2}{3}*( \frac{2}{3})^x+ \frac{2}{3}* \frac{1}{( \frac{2}{3})^x }= \frac{13}{9}$. Substituim pe $(\frac{2}{3})^x=y$, impartim ecuatia cu $\frac{2}{3}$, se obtine ec. $y+ \frac{1}{y}= \frac{13}{6},adica,y^{2}- \frac{13}{6}y+1=0$, avem Δ=($( \frac{13}{6})^2-4= \frac{25}{36},$, radacinile sunt: y=2/3 si y=3/2, deci $x_{1}=1,si, x_{2}=-1$