# Simplification of rational expressions I have the following expression: <mrow class="MJX-TeXA

Simplification of rational expressions
I have the following expression:
$\frac{2}{x-2}+\frac{2}{{x}^{2}-5x+6}$
So I can simplify this as:
$\frac{2}{x-2}+\frac{2}{\left(x-3\right)\left(x-2\right)}$
I make the common denominator to be $\left(x-3\right)\left(x-2\right)$
So I then apply $\left(x-3\right)$ to the left hand side which gives me:
$\frac{2\left(x-3\right)+2}{\left(x-3\right)\left(x-2\right)}$
I have clearly taken a wrong step because the answer in the book to the original expression is $\frac{2}{x-3}$ so I'm not sure how that answer was arrived at.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

zlepljalz2
You did nothing wrong. Note that the numerator can be written as
$2\left(x-3\right)+2=2x-6+2=2x-4=2\left(x-2\right).$

aangenaamyj
Notice, the following steps
$\frac{2}{x-2}+\frac{2}{{x}^{2}-5x+6}$
$=2\left(\frac{1}{x-2}+\frac{1}{\left(x-2\right)\left(x-3\right)}\right)$
$=\frac{2}{x-2}\left(1+\frac{1}{x-3}\right)$
$=\frac{2}{x-2}\left(\frac{x-3+1}{x-3}\right)$
$=\frac{2}{x-2}\left(\frac{x-2}{x-3}\right)$
$=\frac{2}{x-3}$
Your book has the correct expression.