Question

2^(x+1)-11+15/(2^x+1)=0

Answer 1

   
[tex]\displaystyle \\ 2^{x+1}-11+ \frac{15}{2^x+1}=0~~~~~~~| \times (2^x+1) \\ \\ 2^{x+1} \cdot (2^x+1) -11 \cdot (2^x+1)+ 15 =0 \\ 2^{x+1} \cdot 2^x+ 2^{x+1} -11 \cdot 2^x-11+ 15 =0 \\ 2^{2x+1}+ 2^x \cdot 2 -11 \cdot 2^x+4 =0 \\ 2 \cdot 2^{2x} -9 \cdot 2^x+4 =0 \\ \texttt{Substitutie: }~ \boxed{2^x =t } \\ 2t^2 -9t +4=0 \\ \\ t_{12} = \frac{9 \pm \sqrt{81- 32} }{4}=\frac{9 \pm \sqrt{49} }{4}= \frac{9 \pm 7 }{4} [/tex]


[tex]\displaystyle \\t_1 = \frac{9 + 7 }{4} = \frac{16 }{4} = 4 = \boxed{2^2} \\ \\ t_2 = \frac{9 - 7 }{4} = \frac{2 }{4} = \frac{1 }{2} = \boxed{2^{-1}} [/tex]


[tex]\\ \\ \texttt{Ne intoarcem la substitutie: }~~2^x =t \\ \\ \texttt{Solutia 1: } \\ 2^x = t_1 \\ 2^x = 2^2 \\ \Longrightarrow ~~x_1 = \boxed{\boxed{2}} \\ \\ \texttt{Solutia 2: } \\ 2^x = t_2 \\ 2^x = 2^{-1} \\ \Longrightarrow ~~x_2 = \boxed{\boxed{-1 }} [/tex]