Question

In the following exercises, perform indicated operations.

$\frac{3}{x^2-3x-4}-\frac{2}{x^2-5x+4}$

Short answer

The required LCD is$\frac{x-5}{\left(x-4\right)\left(x+1\right)\left(x-1\right)}$.

Step-by-step solution

Step 1. Given Information

The given expression is $\frac{3}{x^2-3x-4}-\frac{2}{x^2-5x+4}$.

Step 2. Find LCD

• Factor the denominators of the given rational expressions.

$$\begin{gathered} x^2-3x-4=\left(x-4\right)\left(x+1\right) \\ {x}^2-5x+4=\left(x-4\right)\left(x-1\right) \end{gathered}$$

• So, the LCD is$\left(x-4\right)\left(x+1\right)\left(x-1\right)$

Step 3. Express with LCD

• Express the expression with LCD.

$\frac{3}{x^2-3x-4}-\frac{2}{x^2-5x+4}=\frac{3x-3}{\left(x-4\right)\left(x+1\right)\left(x-1\right)}-\frac{2x+2}{\left(x-4\right)\left(x+1\right)\left(x-1\right)}$

Step 4. Simplify

• As the denominator is the same for both the terms, perform subtraction in the numerator and place it over the common denominator.

$\frac{3x-3-2x-2}{\left(x-4\right)\left(x+1\right)\left(x-1\right)}=\frac{x-5}{\left(x-4\right)\left(x+1\right)\left(x-1\right)}$

• So, the required expression is$\frac{x-5}{\left(x-4\right)\left(x+1\right)\left(x-1\right)}$.