# How do you add or subtract 2/(x^2+8x+15) + 1/(x^2+11x+30)?

Apr 17, 2018

$\frac{3}{\left(x + 3\right) \left(x + 6\right)}$

#### Explanation:

factorise both denominators:

${x}^{2} + 8 x + 15 = \left(x + 3\right) \left(x + 5\right)$

${x}^{2} + 11 x + 30 = \left(x + 6\right) \left(x + 5\right)$

then the fractions can be written as $\frac{2}{\left(x + 3\right) \left(x + 5\right)}$ and $\frac{1}{\left(x + 6\right) \left(x + 5\right)}$.

the common denominator between these is their lowest common multiple.

they have a common factor of $x + 6$, and the lowest common multiple of $x + 3$ and $x + 5$ is $\left(x + 3\right) \left(x + 5\right)$.

therefore the lowest common multiple of $\left(x + 3\right) \left(x + 5\right)$ and $\left(x + 6\right) \left(x + 5\right)$ is $\left(x + 3\right) \left(x + 6\right) \left(x + 5\right)$.

$\frac{2}{\left(x + 3\right) \left(x + 5\right)} - \frac{2 \left(x + 6\right)}{\left(x + 3\right) \left(x + 6\right) \left(x + 5\right)}$

$= \frac{2 x + 12}{\left(x + 3\right) \left(x + 6\right) \left(x + 5\right)}$

$\frac{1}{\left(x + 6\right) \left(x + 5\right)} = \frac{x + 3}{\left(x + 3\right) \left(x + 6\right) \left(x + 5\right)}$

$2 x + 12 + x + 3 = 3 x + 15$

$3 x + 15 = 3 \left(x + 5\right)$

$3 \frac{x + 5}{\left(x + 3\right) \left(x + 6\right) \left(x + 5\right)} = \frac{3}{\left(x + 3\right) \left(x + 6\right)}$