How do you simplify (18x)/(x^2-5x-36) + (2x)/(x+4)?

Oct 10, 2015

$\frac{2 {x}^{2}}{{x}^{2} - 5 x - 36}$

Explanation:

Lets start by taking a look at the denominator of the first fraction,

${x}^{2} - 5 x - 36$

It turns out that we can factor this expression to get;

$\left(x - 9\right) \left(x + 4\right)$

This means that we have a common term, $\left(x + 4\right)$, in the denominator of both fractions. To get a common denominator, we just need to multiply the second fraction by $\frac{x - 9}{x - 9}$.

$\frac{18 x}{\left(x - 9\right) \left(x + 4\right)} + \frac{2 x}{\left(x + 4\right)} \cdot \textcolor{red}{\frac{\left(x - 9\right)}{\left(x - 9\right)}}$

Now that we have a common denominator, we can combine the two terms into one fraction;

$\frac{18 x + 2 x \left(x - 9\right)}{\left(x - 9\right) \left(x + 4\right)}$

Foiling the denominator, we get back ${x}^{2} - 5 x - 36$ and multiplying the $2 x$ in the numerator through $x - 9$ we get;

$\frac{\cancel{18 x} + 2 {x}^{2} - \cancel{18 x}}{{x}^{2} - 5 x - 36} = \frac{2 {x}^{2}}{{x}^{2} - 5 x - 36}$