Question

sa se rezolve ecuatia:(am nevoie urgent)
9 la puterea(2x+1)- 3 la puterea(x+1)-6=0

Answer 1

[tex]9^{2x+1} -3^{2x+1}-6=0 \Leftrightarrow (3^2)^{2x+1} -3^{2x+1}-6=0 \\\;\\ \Leftrightarrow (3^{2x+1})^2 -3^{2x+1}-6=0 \\\;\\ Notam\ 3^{2x+1}=t, \ t\ \textgreater \ 0,\ iar\ ecuatia\ devine: \\\;\\ t^2-t-6=0, cu \ solutiile: \\\;\\ t_1 = - 2 \ (nu\ convine), \ t_2=3 \\\;\\ t=3 \ \Longrightarrow 3^{2x+1} =3 \Longrightarrow 2x+1=1 \Longrightarrow x=0[/tex]

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Answer 2

   
[tex]9^{(2x+1)}- 3^{(x+1)}-6=0 \\ 9^{(2x+1)}- 3^{(x+1)}=6 \\ (3^2)^{(2x+1)}- 3^{(x+1)}=6 \\ 3^{2\cdot (2x+1)}- 3^{(x+1)}=6 \\ 3^{(4x+2)}- 3^{(x+1)}=6 ~~~\text{Dam factor comun pe: } 3^{(x+1)}\\ 3^{(x+1)}( 3^{(4x+2)-(x+1)} -1 ) = 6 \\ 3^{(x+1)}( 3^{(4x+2-x-1)} -1 ) = 6 \\ 3^{(x+1)}( 3^{(3x+1)} -1 ) = 6 \\ [/tex]


[tex]\texttt{Avem un produs de 2 termeni. } \\ \texttt{Il descompunem pe 6: } \\ 6 = 2 \times 3~~~~~sau~~~~~6 = 1 \times 6 \\ 3^{(x+1)} = 3~~~~~\texttt{Nu poate fi nici 2, nici 1, nici 6} \\ 3^{(3x+1)} -1=2~~~~~\texttt{Nu poate fi nici 3, nici 1, nici 6} \\ \\ \texttt{Rezolvam ecuatiile: } \\ \\ 3^{(x+1)} = 3 ~~\Longrightarrow~~3^{(x+1)} = 3^1 ~~\Longrightarrow~~ x+1=1 ~~\Longrightarrow~~x = 1-1=\boxed{0} [/tex]


[tex]3^{(3x+1)} -1=2 ~~\Longrightarrow~~3^{(3x+1)}=2+1 \\ \Longrightarrow~~3^{(3x+1)}=3\\ \Longrightarrow~~3^{(3x+1)}=3^1\\ \Longrightarrow~~3x+1=1 ~~\Longrightarrow~~ x = \frac{1-1}{3}= \frac{0}{3} = \boxed{0}\\ \\ \texttt{Solutia problemei: } \boxed{x = 0 } ~~\texttt{Solutie unica.}[/tex]