Question

10(elevado a 2x-1)-11.10(elevado a x -1)+1=0

Answer 1

Olá,

use as propriedades da exponenciação, já estudadas..

$10^{2x-1}-11\cdot10^{x-1}+1=0\\ 10^{2x}\cdot10^{-1}-11\cdot10^x\cdot10^{-1}+1=0\\\\ \dfrac{1}{10}\cdot (10^x)^2-11\cdot \dfrac{1}{10}\cdot10^x+1=0\\\\ \dfrac{1}{10}\cdot(10^x)^2- \dfrac{11}{10}\cdot10^x+1=0\\\\ 10^x=y\\\\ \dfrac{1}{10}y^2- \dfrac{11}{10}y+1=0\\\\ \dfrac{1}{10} y^2- \dfrac{11}{10}y+ \dfrac{10}{10}=0\\\\ y^2-11y+10=0$

$\Delta=(-11)^2-4\cdot1\cdot10\\ \Delta=121-40\\ \Delta=81\\\\ y= \dfrac{-(-11)\pm \sqrt{81} }{2\cdot1}= \dfrac{11\pm9}{2}\begin{cases}y_1=1\\ y_2=10\end{cases}\\\\\\ 10^x=y\\\\\\ 10^x=1~~~~~~~~~10^x=10\\ \not10^x=\not10^0~~~~~\not10^x=\not10^1\\\\ x_1=0~~~~~~~~~~~~x_2=1$

$\huge\boxed{\text{S}=\{0,1\}}$