Simplify ((a^2-14a+49)/(a^2-a-42))÷((a^2-49)/7)

Rewrite ${\displaystyle 49}$ as ${\displaystyle 7^2}$.

Check the middle term by multiplying ${\displaystyle 2ab}$ and compare this result with the middle term in the original expression.

Simplify.

Factor using the perfect square trinomial rule ${\displaystyle a^2-2ab+b^2={(a-b)}^2}$, where ${\displaystyle a=a}$ and ${\displaystyle b=-7}$.

Consider the form ${\displaystyle x^2+bx+c}$. Find a pair of integers whose product is ${\displaystyle c}$ and whose sum is ${\displaystyle b}$. In this case, whose product is ${\displaystyle -42}$ and whose sum is ${\displaystyle -1}$.

Write the factored form using these integers.

Factor ${\displaystyle a-7}$ out of ${\displaystyle {(a-7)}^2}$.

Cancel the common factors.

Rewrite ${\displaystyle 49}$ as ${\displaystyle 7^2}$.

Since both terms are perfect squares, factor using the difference of squares formula, ${\displaystyle a^2-b^2=(a+b)(a-b)}$ where ${\displaystyle a=a}$ and ${\displaystyle b=7}$.

Cancel the common factor of ${\displaystyle a-7}$.

Multiply ${\displaystyle \frac{1}{a+6}}$ and ${\displaystyle \frac{7}{a+7}}$.