Задание 53

Question

Упростите выражение:
$$\left(\frac{m}{m^2 - 49} - \frac{m + 7}{m^2 - 5m - 14}\right) \cdot \frac{m^2 + 9m + 14}{-12m - 49}.$$

Gimli · Accepted Answer

\begin{align*}
&\left(\frac{m}{m^2 - 49} - \frac{m + 7}{m^2 - 5m - 14}\right) \cdot \frac{m^2 + 9m + 14}{-12m - 49}\[10pt]

&= \left(\frac{m}{(m - 7)(m + 7)} - \frac{m +7}{(m - 7)(m + 2)}\right)\cdot \frac{m^2 + 9m + 14}{-12m - 49}\[10pt]

&= \frac{m(m + 2) - (m + 7)^2}{(m^2 + 9m + 14)(m - 7)}\cdot \frac{m^2 + 9m + 14}{-12m - 49}\[10pt]

&= \frac{m^2 + 2m - m^2 - 14m - 49}{(m^2 + 9m + 14)(m - 7)}\cdot \frac{m^2 -5m - 14}{-12m - 49}\[10pt]

&= \frac{-12m - 49}{(m^2 + 9m + 14)(m - 7)}\cdot \frac{m^2 + 9m + 14}{-12m - 49}\[10pt]

&= \frac{1}{m - 7}.
\end{align*}

\textbf{Ответ: $\frac{1}{m - 7}$.}

Answers