Question

resolva a equação exponencial 3^{2x} - 4 . 3^{x} + 3=0

Answer 1

Oi Thais,

vamos fazer uso de uma variável auxiliar, 3^x=y:

$3^{2x}-4\cdot3^x+3=0~~.\\ (3^x)^2-4\cdot3^x+3=0\\\\ 3^x=y\\\\ y^2-4y+3=0\\ \Delta=b^2-4ac\\ \Delta=(-4)^2-4\cdot1\cdot3\\ \Delta=16-12\\ \Delta=4\\\\ y= \dfrac{-b\pm \sqrt{\Delta} }{2a}= \dfrac{-(-4)\pm \sqrt{4} }{2\cdot1}= \dfrac{4\pm2}{2}\begin{cases}y'= \dfrac{4-2}{2}= \dfrac{2}{2}=1\\\\ y''= \dfrac{4+2}{2}= \dfrac{6}{2}=3 \end{cases}$

Retomando a variável original, 3^x=y:

$3^x=1~~~~~~~~~~~~~~~~~~~3^x=3\\ 3^x=3^0~~~~~~~~~~~~~~~~~~3^x=3^1\\ \not3^x=\not3^0~~~~~~~~~~~~~~~~~\not3^x=\not3^1\\\\ x=0~~~~~~~~~~~~~~~~~~~~~~x=1\\\\\\ \Large\boxed{\boxed{\boxed{S=\{0,1\}}}}.\\.$

Tenha ótimos estudos ^^