Question

o conjunto solução da equação log12 (x2-x)=1

Answer 1

Pela definição y = log_{a}(b) ⇔ a^{y} = b

$\therefore~~\ell og_{12}(x^2-x)=1~\Leftrightarrow~12^1=x^2-x$

Vamos resolver a equação quadrática pelo método de completar quadrados:

$x^2-x=12\\\\x^2-x+(\frac{1}{2})^2=12+(\frac{1}{2})^2\\\\(x-\frac{1}{2})^2=12+\frac{1}{4}\\\\(x-\frac{1}{2})^2=\frac{48}{4}+\frac{1}{4}\\\\(x-\frac{1}{2})^2=\frac{49}{4}\\\\\sqrt{(x-\frac{1}{2})^2}=\sqrt{\frac{49}{4}}\\\\|x-\frac{1}{2}|=\frac{7}{2}$


$\begin{Bmatrix}x-\frac{1}{2}=-\frac{7}{2}~\rightarrow~x=-\frac{7}{2}+\frac{1}{2}~\rightarrow~x=-\frac{6}{2}~\rightarrow~x=-3\\\\ou\\\\x-\frac{1}{2}=\frac{7}{2}~\rightarrow~x=\frac{7}{2}+\frac{1}{2}~\rightarrow~x=\frac{8}{2}~\rightarrow~x=4~~~~~~~~~\end.$


$\large\fbox{ S=\begin{Bmatrix}x=-3~~ou~~x=4\end{Bmatrix} }~~\leftarrow~\text{resposta}$