Question

what is this expression in simplest form?
\\(\frac{4w}{w - 2} + \frac{3w}{w - 3}\\)
\\(\bigcirc\\) a. \\(\frac{7w}{2w - 5}\\)
\\(\bigcirc\\) b. \\(\frac{7w}{w^2 + 6}\\)
\\(\bigcirc\\) c. \\(\frac{7w^2 - 18w}{w^2 - 5w + 6}\\)
\\(\bigcirc\\) d. \\(\frac{7w^2 - 6}{w^2 - 5w + 6}\\)

Explanation:

Step1: Find a common denominator

The denominators are \( w - 2 \) and \( w - 3 \), so the common denominator is \( (w - 2)(w - 3) \).

Step2: Rewrite each fraction with the common denominator

\( \frac{4w}{w - 2} = \frac{4w(w - 3)}{(w - 2)(w - 3)} \) and \( \frac{3w}{w - 3} = \frac{3w(w - 2)}{(w - 2)(w - 3)} \)

Step3: Add the numerators

\( \frac{4w(w - 3) + 3w(w - 2)}{(w - 2)(w - 3)} \)

Step4: Expand the numerators

\( 4w(w - 3) = 4w^2 - 12w \) and \( 3w(w - 2) = 3w^2 - 6w \)

Step5: Combine like terms in the numerator

\( 4w^2 - 12w + 3w^2 - 6w = 7w^2 - 18w \)

Step6: Expand the denominator

\( (w - 2)(w - 3) = w^2 - 5w + 6 \)

Answer:

C. \( \frac{7w^2 - 18w}{w^2 - 5w + 6} \)