 # Simplification of rational expressions I have the following expression: <mrow class="MJX-TeXA Augustus Acevedo 2022-07-07 Answered
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.
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You did nothing wrong. Note that the numerator can be written as
$2\left(x-3\right)+2=2x-6+2=2x-4=2\left(x-2\right).$

We have step-by-step solutions for your answer! 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.

We have step-by-step solutions for your answer!