Question

Express the difference below in the simplest form $?$

$\frac{9}{x^2-2x-15}-\frac{2}{x+3}$

Answer 1

Find the difference in simplest form:

$\frac{9}{{\mathrm{x}}^2-2\mathrm{x}-15}-\frac{2}{\mathrm{x}+3}$

Use splitting the middle term to factorize ${\mathrm{x}}^2-2\mathrm{x}-15$:

(Take two numbers such that their sum is $-2$ and their product is $-15$)

$=x^2-5x+3x-15$

$=\mathrm{x}(\mathrm{x}-5)+3(\mathrm{x}-5)$ (take common terms)

$=(\mathrm{x}-5)(\mathrm{x}+3)$

Now, the expression will be:

$\frac{9}{(\mathrm{x}-5)(\mathrm{x}+3)}-\frac{2}{\mathrm{x}+3}$

$$\begin{gathered} =\frac{9-2(\mathrm{x}-5)}{(\mathrm{x}-5)(\mathrm{x}+3)} \\ =\frac{9-2\mathrm{x}+10}{(\mathrm{x}-5)(\mathrm{x}+3)} \end{gathered}$$

(add like terms together)

$=\frac{-2\mathrm{x}+19}{(\mathrm{x}-5)(\mathrm{x}+3)}$

Hence, the difference in simplest form is $\frac{\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{19}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{5}\mathbf{)}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{)}}$.