Question

O conjunto solução da equação exponencial (3^x)^2 - 12. 3^x + 27 = 0 é:

Answer 1

Resposta:

$\textsf{Leia abaixo}$

Explicação passo a passo:

$\mathsf{(3^x)^2 - 12.3^x + 27 = 0}$

$\mathsf{y = 3^x}$

$\mathsf{y^2 - 12y + 27 = 0}$

$\mathsf{\Delta = b^2 - 4.a.c}$

$\mathsf{\Delta = (-12)^2 - 4.1.27}$

$\mathsf{\Delta = 144 - 108}$

$\mathsf{\Delta = 36}$

$\mathsf{y = \dfrac{-b \pm \sqrt{\Delta}}{2a} = \dfrac{12 \pm \sqrt{36}}{2} \rightarrow \begin{cases}\mathsf{y' = \dfrac{12 + 6}{2} = \dfrac{18}{2} = 9}\\\\\mathsf{y'' = \dfrac{12 - 6}{2} = \dfrac{6}{2} = 3}\end{cases}}$

$\mathsf{3^x = 9 \iff \not3^x = \not3^2 \iff x = 2}$

$\mathsf{3^x = 3 \iff \not3^x = \not3^1 \iff x = 1}$

$\boxed{\boxed{\mathsf{S = \{2;1\}}}}$