Gente me ajudem a resolver essa questão, eu to meio confusa: 25(elevado a x) = 6 x 5(elevado a x)

Biancasauer2001 @Biancasauer2001

January 2020 1 52 Report

Gente me ajudem a resolver essa questão, eu to meio confusa:
25(elevado a x) = 6 x 5(elevado a x) - 5

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korvo Oi Bianca,

use uma das propriedades da exponenciação, acompanhe a resolução:

$\mathsf{25^x=6\cdot5^x-5}\\
\mathsf{(5^2)^x-6\cdot5^x+5=0}\\
\mathsf{(5^x)^2-6\cdot5^x+5=0}\\\\
\mathsf{5^x=y}\\\\
\mathsf{y^2-6y+5=0}\\\\
\mathsf{\Delta=(-6)^2-4\cdot1\cdot5}\\
\mathsf{\Delta=36-20}\\
\mathsf{\Delta=16}\\\\
\mathsf{y= \dfrac{-(-6)\pm \sqrt{16} }{2\cdot1} = \dfrac{6\pm4}{2} }\begin{cases}\mathsf{y_1= \dfrac{6-4}{2}= \dfrac{2}{2}=1 }\\\\
\mathsf{y_2= \dfrac{6+4}{2} = \dfrac{10}{2} =5}\end{cases}$

$\mathsf{5^x=y, ~Lembra?}\\\\
\mathsf{5^x=1}~~~~~~~~~~~~~~~~~~\mathsf{5^x=5}\\
\mathsf{5^x=5^o}~~~~~~~~~~~~~~~~~\mathsf{5^x=5^1}\\
\mathsf{\not5^x=\not5^o}~~~~~~~~~~~~~~~\mathsf{\not5^x=\not5^1}\\\\
\mathsf{~~x_1=0}~~~~~~~~~~~~~~~~~~\mathsf{x_2=1}\\\\\\
\Large\boxed{\mathsf{S=\{0,1\}}}$

Se tiver dúvidas me chame. Tenha ótimos estudos! :))

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