Question

Rezolvati ecuatiile ( exponentiale )

(2+$\sqrt{3}$)^x - (2-$\sqrt{3}$)^x = 2√3

Answer 1

notam
$(2+ \sqrt{3} )^{x}=t;t\ \textgreater \ 0 \\ t- \frac{1}{t} =2 \sqrt{3} \\ t^{2}-2 \sqrt{3}t-1=0$
Δ=12+4=16
$t_{1}= \frac{2 \sqrt{3}-4 }{2}= \sqrt{3}-2\ \textless \ 0 \\ t_{2}= \frac{2 \sqrt{3}+4 }{2}=2+ \sqrt{3}$
revenim la substitutie
$(2+ \sqrt{3})^x =2+ \sqrt{3} \\ x=1$