Question

find the difference.
\\(\dfrac{2g}{g^2 - g - 6} - \dfrac{1}{g - 3}\\)
\\(\text{a. } \dfrac{1}{g - 3}\\)
\\(\text{b. } \dfrac{g - 2}{g^2 - g - 6}\\)
\\(\text{c. } \dfrac{2g - 1}{g^2 - 2g - 3}\\)
\\(\text{d. } \dfrac{g}{g + 2}\\)

Explanation:

Step1: Factor the denominator

Factor \(g^2 - g - 6\). We need two numbers that multiply to \(-6\) and add to \(-1\). Those numbers are \(-3\) and \(2\). So, \(g^2 - g - 6=(g - 3)(g + 2)\).
The first fraction becomes \(\frac{2g}{(g - 3)(g + 2)}\), and the second fraction is \(\frac{1}{g - 3}\).

Step2: Find a common denominator

The common denominator of \((g - 3)(g + 2)\) and \(g - 3\) is \((g - 3)(g + 2)\). Rewrite the second fraction with the common denominator: \(\frac{1}{g - 3}=\frac{g + 2}{(g - 3)(g + 2)}\).

Step3: Subtract the fractions

Now subtract the two fractions: \(\frac{2g}{(g - 3)(g + 2)}-\frac{g + 2}{(g - 3)(g + 2)}=\frac{2g-(g + 2)}{(g - 3)(g + 2)}\).

Step4: Simplify the numerator

Simplify the numerator: \(2g-(g + 2)=2g - g - 2=g - 2\). So the result is \(\frac{g - 2}{(g - 3)(g + 2)}\), and since \((g - 3)(g + 2)=g^2 - g - 6\), the fraction is \(\frac{g - 2}{g^2 - g - 6}\).

Answer:

B. \(\dfrac{g - 2}{g^2 - g - 6}\)