Question

$\frac{2v}{3v^{2}-21v + 36}-\frac{3}{2v}$
$\frac{av^{2}+bv + c}{6v(v - 4)(v - 3)}$
$a=$
$b=$
$c=$

Explanation:

Step1: Factor the denominator

First, factor $3v^{2}-21v + 36=3(v^{2}-7v + 12)=3(v - 3)(v - 4)$.

Step2: Find a common - denominator

The common denominator of $\frac{2v}{3(v - 3)(v - 4)}$ and $\frac{3}{2v}$ is $6v(v - 3)(v - 4)$.
Rewrite $\frac{2v}{3(v - 3)(v - 4)}$ as $\frac{2v\times2v}{6v(v - 3)(v - 4)}=\frac{4v^{2}}{6v(v - 3)(v - 4)}$ and $\frac{3}{2v}$ as $\frac{3\times3(v - 3)(v - 4)}{6v(v - 3)(v - 4)}=\frac{9(v^{2}-7v + 12)}{6v(v - 3)(v - 4)}=\frac{9v^{2}-63v + 108}{6v(v - 3)(v - 4)}$.

Step3: Subtract the fractions

$\frac{4v^{2}}{6v(v - 3)(v - 4)}-\frac{9v^{2}-63v + 108}{6v(v - 3)(v - 4)}=\frac{4v^{2}-(9v^{2}-63v + 108)}{6v(v - 3)(v - 4)}=\frac{4v^{2}-9v^{2}+63v - 108}{6v(v - 3)(v - 4)}=\frac{- 5v^{2}+63v - 108}{6v(v - 3)(v - 4)}$.
Since $\frac{-5v^{2}+63v - 108}{6v(v - 3)(v - 4)}=\frac{Av^{2}+Bv + C}{6v(v - 3)(v - 4)}$, then $A=-5$, $B = 63$, $C=-108$.

Answer:

$A=-5$
$B = 63$
$C=-108$