Question

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

Explanation:

Step1: Find a common denominator

The denominators are \(w - 2\) and \(w - 3\), so the common denominator is \((w - 2)(w - 3)=w^{2}-5w + 6\).

Step2: Rewrite each fraction with the common denominator

For \(\frac{4w}{w - 2}\), multiply numerator and denominator by \(w - 3\): \(\frac{4w(w - 3)}{(w - 2)(w - 3)}=\frac{4w^{2}-12w}{w^{2}-5w + 6}\).
For \(\frac{3w}{w - 3}\), multiply numerator and denominator by \(w - 2\): \(\frac{3w(w - 2)}{(w - 3)(w - 2)}=\frac{3w^{2}-6w}{w^{2}-5w + 6}\).

Step3: Add the two fractions

\(\frac{4w^{2}-12w}{w^{2}-5w + 6}+\frac{3w^{2}-6w}{w^{2}-5w + 6}=\frac{(4w^{2}-12w)+(3w^{2}-6w)}{w^{2}-5w + 6}\).
Combine like terms in the numerator: \(4w^{2}+3w^{2}=7w^{2}\), \(-12w-6w=-18w\), so the numerator is \(7w^{2}-18w\).

Answer:

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