Answer
$\frac{x+25}{2(x-5)(x-4)}$.
Work Step by Step
In order to determine the Least Common Denominator (LCD) for the fractions in the given expression, we first factor all denominators.
Factor $x^2-9x+20$
Rewrite the middle term $-9x$ as $-5x-4x$.
$\Rightarrow x^2-5x-4x+20$
Group terms.
$\Rightarrow (x^2-5x)+(-4x+20)$
Factor each group.
$\Rightarrow x(x-5)-4(x-5)$
Factor out $(x-5)$.
$\Rightarrow (x-5)(x-4)$
Factor $2x-8$
Factor out $2$.
$\Rightarrow 2(x-4)$
Substitute back all factors into the given expression.
$\Rightarrow\frac{3x}{x^2-9x+20}-\frac{5}{2x-8}= \frac{3x}{(x-5)(x-4)}-\frac{5}{2(x-4)}$
The LCD is $2(x-5)(x-4)$.
Multiply the numerator and the denominator to form LCD at the denominators.
$\Rightarrow \frac{2\cdot3x}{2(x-5)(x-4)}-\frac{5(x-5)}{2(x-5)(x-4)}$
Use the distributive property.
$\Rightarrow \frac{6x}{2(x-5)(x-4)}-\frac{5x-25}{2(x-5)(x-4)}$
Add numerators because denominators are the same.
$\Rightarrow \frac{6x-(5x-25)}{2(x-5)(x-4)}$
Simplify.
$\Rightarrow \frac{6x-5x+25}{2(x-5)(x-4)}$
$\Rightarrow \frac{x+25}{2(x-5)(x-4)}$.
