Question

Suma solutiilor ecuatiei $9^{x} -5*3^{x}+6=0$ este ?

Answer 1

fie
[tex] 3^{x}=t,t\ \textgreater \ 0 \\ t^{2} -5t+6=0 \\ t=2;t=3 \\ 3^{x}=3;x=1 \\ 3^{x}=2;x= log_{3}2 \\ S=1+ log_{3}2= log_{3}6 [/tex]

Answer 2

$\displaystyle \mathtt{9^x-5 \cdot 3^x+6=0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3^x=t}\\ \\ \mathtt{t^2-5t+6=0} \\ \\ \mathtt{a=1,~b=-5,~c=6}\\ \\ \mathtt{\Delta=b^2-4ac=(-5)^2-4 \cdot 1 \cdot 6=25-24=1\ \textgreater \ 0}\\ \\ \mathtt{t_1= \frac{-b+ \sqrt{\Delta} }{2a} = \frac{-(-5)+ \sqrt{1} }{2 \cdot 1} = \frac{5+1}{2}= \frac{6}{2}=3}\\ \\ \mathtt{t_2= \frac{-b- \sqrt{\Delta} }{2a} = \frac{-(-5)- \sqrt{1} }{2 \cdot 1}= \frac{5-1}{2}= \frac{4}{2} =2 }$
$\displaystyle \mathtt{3^x=3 \Rightarrow x=1;~~~~~~~~~~~~~~~~~~~~~~~~3^x=2 \Rightarrow x=log_32} \\ \\ \mathtt{S=1+log_32=log_33+log_32=log_3(3 \cdot 2)=log_36}$