Question

(Uniube-MG)A soma das raízes da equação:
4.3^|5x-2| - 9.^|5x-2|=3
Resposta 6/5

Answer 1

$4\cdot3^{|5x-2|}-9^{|5x-2|}=3\\\\ 4\cdot3^{|5x-2|}-(3^2)^{|5x-2|}=3\\\\ 4\cdot3^{|5x-2|}-3^{2|5x-2|}=3\\\\ 4\cdot3^{|5x-2|}-(3^{|5x-2|})^2=3$

Vamos fazer uma substituição. Seja y tal que $y=3^{|5x-2|}$. Usando na equação acima:

$4y-y^2=3\\\\ y^2-4y+3=0\\\\\\ \Delta=b^2-4ac\\\\ \Delta=(-4)^2-4\cdot1\cdot3\\\\ \Delta=16-12\\\\ \Delta=4\\\\\\ y=\dfrac{-b\pm\sqrt\Delta}{2a}\\\\ y=\dfrac{-(-4)\pm\sqrt4}{2\cdot1}\\\\ y=\dfrac{4\pm2}{2}\\\\ y=2\pm1\\\\ \begin{cases}y=2+1\iff\boxed{y=3}\\y=2-1\iff\boxed{y=1}\end{cases}$

Desfazendo a substituição para cada valor de y obtido:

$\text{Para y=3:}\\\\ 3^{|5x-2|}=3\\\\ 3^{|5x-2|}=3^1\\\\ |5x-2|=1\\\\ \begin{cases}5x-2=1\iff5x=3\iff\boxed{x=\frac{3}{5}}\\ 5x-2=-1\iff 5x=1\iff\boxed{x=\frac{1}{5}}\end{cases}$

$\\\\ \text{Para y=1:}\\\\ 3^{|5x-2|}=1\\\\ 3^{|5x-2|}=3^0\\\\ |5x-2|=0\\\\ 5x-2=0\\\\ 5x=2\\\\ \boxed{x=\dfrac{2}{5}}$

Obtendo a soma S das raízes encontradas acima:

$S=\dfrac{3}{5}+\dfrac{1}{5}+\dfrac{2}{5}\\\\ \boxed{\boxed{S=\dfrac{6}{5}}}\to\text{Letra E}$