Question

Express the following difference in the simplest form $?$

$\frac{5}{\mathrm{x}+5}-\frac{2\mathrm{x}+5}{{\mathrm{x}}^2+9\mathrm{x}+20}$

Answer 1

Find the expression in the simplest form:

$\frac{5}{\mathrm{x}+5}-\frac{2\mathrm{x}+5}{{\mathrm{x}}^2+9\mathrm{x}+20}$

Use splitting the middle term to factorize ${\mathrm{x}}^2+9\mathrm{x}+20$:

(Take two number in such a way that their sum is $9$ and their product is $20$)

$$\begin{gathered} {\mathrm{x}}^2+5\mathrm{x}+4\mathrm{x}+20 \\ =\mathrm{x}(\mathrm{x}+5)+4(\mathrm{x}+5) \end{gathered}$$

(take the common terms)

$\Rightarrow (\mathrm{x}+5)(\mathrm{x}+4)$

Substitute them in the given expression:

$$\begin{gathered} \frac{5}{\mathrm{x}+5}-\frac{2\mathrm{x}+5}{(\mathrm{x}+5)(\mathrm{x}+4)} \\ =\frac{5(\mathrm{x}+4)-(2\mathrm{x}+5)}{(\mathrm{x}+5)(\mathrm{x}+4)} \\ =\frac{5\mathrm{x}+20-2\mathrm{x}-5}{(\mathrm{x}+5)(\mathrm{x}+4)} \end{gathered}$$

(simplify by grouping the like terms together)

$$\begin{gathered} =\frac{5x-2x+20-5}{(x+5)(x+4)} \\ =\frac{3x+15}{(x+5)(x+4)} \end{gathered}$$

(take the common term in numerator)

$\Rightarrow \frac{3(\mathrm{x}+5)}{(\mathrm{x}+5)(\mathrm{x}+4)}$ (cancel the like terms)

$=\frac{3}{\mathrm{x}+4}$

Hence, the expression in the simplest form is $\frac{\mathbf{3}}{\mathbf{x}\mathbf{+}\mathbf{4}}$.