Question

UFPR
As raízes da equação
$log_{2}( {x}^{2} - 3x + 2) - log_{ \frac{1}{2} }(x - 2) = log_{2}(6) + log_{2}( {x}^{2} - 5x + 6)$
têm soma igual a:
a. 5
b. 9
c. 7
d. 8
e. 6​

Answer 1

Resposta:

$\textsf{Leia abaixo}$

Explicação passo a passo:

$\mathsf{log_2\:(x^2 - 3x + 2) - log_{\frac{1}{2}}\:(x - 2) = log_2\:6 + log_2\:(x^2 - 5x + 6)}$

$\mathsf{log_2\:(x^2 - 3x + 2) - \dfrac{log_{2}\:(x - 2)}{log_2\:1/2} = log_2\:6 + log_2\:(x^2 - 5x + 6)}$

$\mathsf{log_2\:(x^2 - 3x + 2) - \dfrac{log_{2}\:(x - 2)}{log_2\:2^{-1}} = log_2\:6 + log_2\:(x^2 - 5x + 6)}$

$\mathsf{log_2\:(x^2 - 3x + 2) + log_{2}\:(x - 2) = log_2\:6 + log_2\:(x^2 - 5x + 6)}$

$\mathsf{log_2\:(x^2 - 3x + 2).(x - 2) = log_2\:6.(x - 2).(x - 3)}$

$\mathsf{(x^2 - 3x + 2) = 6(x - 3)}$

$\mathsf{x^2 - 3x + 2 = 6x - 18}$

$\mathsf{x^2 - 9x + 20 = 0}$

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

$\mathsf{\Delta = (-9)^2 - 4.1.20}$

$\mathsf{\Delta = 81 - 80}$

$\mathsf{\Delta = 1}$

$\mathsf{x = \dfrac{-b \pm \sqrt{\Delta}}{2a} = \dfrac{9 \pm \sqrt{1}}{2} \rightarrow \begin{cases}\mathsf{x' = \dfrac{9 + 1}{2} = \dfrac{10}{2} = 5}\\\\\mathsf{x'' = \dfrac{9 - 1}{2} = \dfrac{8}{2} = 4}\end{cases}}$

$\boxed{\boxed{\mathsf{S = \{5;4\}}}}\leftarrow\textsf{letra B}$