### Video Transcript

Given that 𝑛 of 𝑥 is equal to
𝑥 squared plus 12𝑥 plus 36 over 𝑥 squared minus 𝑎 simplifies to 𝑛 of 𝑥 is
equal to 𝑥 plus six over 𝑥 minus six, what is the value of 𝑎?

We begin by recalling that to
simplify a rational function, we find its domain, factor the numerator and
denominator, and then cancel the shared factors over the domain. In this question, the
simplified function has a linear denominator, whereas the original function has
a quadratic denominator. We must therefore be able to
factor 𝑥 squared minus 𝑎 into two linear factors, and one of these must be 𝑥
minus six. We notice that our original
expression appears to be of the form 𝑥 squared minus 𝑦 squared. And this can be factored using
the difference of two squares into two linear factors: 𝑥 minus 𝑦, 𝑥 plus
𝑦. Since 𝑦 is equal to six, we
have 𝑥 squared minus 𝑎 is equal to 𝑥 minus six multiplied by 𝑥 plus six. And distributing the
parentheses or noting that 𝑎 is equal to 𝑦 squared, the value of 𝑎 is equal
to 36.

We can verify this by
considering the initial expression for 𝑛 of 𝑥. If 𝑎 is equal to 36, 𝑛 of 𝑥
is equal to 𝑥 squared plus 12𝑥 plus 36 over 𝑥 squared minus 36. The numerator can be factored
into two sets of parentheses, where the first term is 𝑥. The second terms in each of the
parentheses must have a product of 36 and a sum of 12. 𝑥 squared plus 12𝑥 plus 36 is
equal to 𝑥 plus six multiplied by 𝑥 plus six or 𝑥 plus six all squared. As we have already factored the
denominator, 𝑛 of 𝑥 is equal to 𝑥 plus six multiplied by 𝑥 plus six over 𝑥
minus six multiplied by 𝑥 plus six. The denominator has zeros at 𝑥
equals positive and negative six. Therefore, the function is
undefined at these values. This means that the domain of
𝑛 of 𝑥 is the set of all real values minus the set containing negative six and
six. As negative six is not
contained in the domain, we can cancel the factor of 𝑥 plus six. And the function 𝑛 of 𝑥
simplifies to 𝑥 plus six over 𝑥 minus six as required. This confirms that the value of
𝑎 is 36.