Question

Rezolvati in multimea numerelor reale ecuatia 2^{2x+1}-3*2^{x}+1=0

Answer 1

$2^{2x+1}-3\cdot 2^{x}+1=0\\ \\ 2^{2x}\cdot 2 - 2\cdot 2^x+1=0\\ \\ 2\cdot (2^x)^2-3\cdot 2^{x}+1=0\\ \\ Notez \ 2^x=t, \ t>0\\ \\ 2\cdot t^2 -3\cdot t+1=0\\ \\ \Delta=(-3)^2-4\cdot 1\cdot 2=9-8=1\\ \\ t_{1, \ 2}=\frac{-(-3)\pm \sqrt{1}}{2\cdot 2}\\ \\ t_{1, \ 2}=\frac{3\pm 1}{4}\\ \\ t_1=\frac{3+1}{4}=\frac{4}{4}=1=2^{0}>0\\ \\ t_2=\frac{3-1}{4}=\frac{1}{2}=2^{-1}>0\\ \\ \Rightarrow 2^x \ in \{ 0,5; \ 1 \}\\ \\ Cazul \  I:\\ \\ 2^x=2^{-1} \Rightarrow x=-1\\ \\ Cazul \ II:\\ \\ 2^x=2^{0} \Rightarrow x=0\\ \\ \boxed{S=\{  0, \ -1 \} }$