Question

1. simplify: \\(\frac{5}{n + 6} - \frac{6n}{2}\\). teks a2.7 (f)

a) \\(\frac{6 - 4n}{n + 2}\\)
b) \\(\frac{-3n^2 - 18n + 5}{n + 6}\\)
c) \\(\frac{4 + n}{-5n + 6}\\)
d) \\(\frac{6 - 6n}{n + 4}\\)

Explanation:

Step1: Find a common denominator

The denominators are \(n + 6\) and \(2\), so the common denominator is \(2(n + 6)\).
Rewrite each fraction with the common denominator:
\(\frac{5}{n + 6}=\frac{5\times2}{2(n + 6)}=\frac{10}{2(n + 6)}\)
\(\frac{6n}{2}=\frac{6n(n + 6)}{2(n + 6)}\)

Step2: Subtract the fractions

\(\frac{5}{n + 6}-\frac{6n}{2}=\frac{10}{2(n + 6)}-\frac{6n(n + 6)}{2(n + 6)}\)
\(=\frac{10 - 6n(n + 6)}{2(n + 6)}\)

Step3: Expand and simplify the numerator

Expand \(6n(n + 6)\): \(6n^2+36n\)
So the numerator is \(10-(6n^2 + 36n)=10-6n^2-36n=-6n^2-36n + 10\)
We can factor out a \(-2\) from the numerator: \(-2(3n^2 + 18n - 5)\)? Wait, no, let's check the options. Wait, maybe I made a mistake in the common denominator. Wait, the options have denominator \(n + 6\), so maybe the common denominator is \(2(n + 6)\), but let's re - do the subtraction:

Wait, \(\frac{5}{n + 6}-\frac{6n}{2}=\frac{5\times2-6n(n + 6)}{2(n + 6)}=\frac{10-6n^2-36n}{2(n + 6)}\)
Factor out a \(2\) from the numerator: \(\frac{2(5 - 3n^2-18n)}{2(n + 6)}\)
Cancel out the \(2\) in the numerator and denominator: \(\frac{-3n^2-18n + 5}{n + 6}\)

Answer:

B. \(\frac{-3n^{2}-18n + 5}{n + 6}\)