Question

write as a single fraction.\\(\frac{4w}{w^2 - 6w + 8} - \frac{2}{w - 4} - \frac{5}{w + 9}\\)simplify your answer as much as possible.

Explanation:

Step1: Factor the quadratic denominator

$w^2 - 6w + 8 = (w-2)(w-4)$

Step2: Find common denominator

Common denominator: $(w-2)(w-4)(w+9)$

Step3: Rewrite each term

$\frac{4w(w+9)}{(w-2)(w-4)(w+9)} - \frac{2(w-2)(w+9)}{(w-2)(w-4)(w+9)} - \frac{5(w-2)(w-4)}{(w-2)(w-4)(w+9)}$

Step4: Expand numerators

$4w(w+9) = 4w^2 + 36w$
$2(w-2)(w+9) = 2(w^2 +7w -18) = 2w^2 +14w -36$
$5(w-2)(w-4) = 5(w^2 -6w +8) = 5w^2 -30w +40$

Step5: Combine numerators

$(4w^2 +36w) - (2w^2 +14w -36) - (5w^2 -30w +40)$
$=4w^2+36w-2w^2-14w+36-5w^2+30w-40$

Step6: Simplify numerator

$(4w^2-2w^2-5w^2)+(36w-14w+30w)+(36-40)$
$=-3w^2 +52w -4$

Answer:

$\frac{-3w^2 + 52w - 4}{(w-2)(w-4)(w+9)}$