Question

In the following exercises, perform indicated operations.

$\frac{4}{x^2-6x+5}-\frac{3}{x^2-7x+10}$

Short answer

The required expression is$\begin{array}{rcl}& & \frac{1}{\left(x-1\right)\left(x-2\right)}\end{array}$

Step-by-step solution

Step 1. Given information

The given expression is $\frac{4}{x^2-6x+5}-\frac{3}{x^2-7x+10}$

Step 2. Find LCD

• Factor the denominators of the given rational expressions.

$\begin{array}{rcl}{x}^2-6x+5& =& \left(x-5\right)\left(x-1\right)\\ x^2-7x+10& =& \left(x-5\right)\left(x-2\right)\end{array}$

• So, the LCD is$\left(x-5\right)\left(x-1\right)\left(x-2\right)$

Step 3. Express with LCD

• Rewrite the expression in terms of LCD.

$\frac{4}{x^2-6x+5}-\frac{3}{x^2-7x+10}=\frac{4x-8}{\left(x-5\right)\left(x-1\right)\left(x-2\right)}-\frac{3x-3}{\left(x-5\right)\left(x-1\right)\left(x-2\right)}$

Stp 4. Simplify

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

$\begin{array}{rcl}\frac{4x-8-3x+3}{\left(x-5\right)\left(x-1\right)\left(x-2\right)}& =& \frac{x-5}{\left(x-5\right)\left(x-1\right)\left(x-2\right)}\\ & =& \frac{1}{\left(x-1\right)\left(x-2\right)}\end{array}$

• So, the required expression is $\begin{array}{rcl}& & \frac{1}{\left(x-1\right)\left(x-2\right)}\end{array}$.