Question

Express the following difference in simplest form $?$

$\frac{7\mathrm{x}}{{\mathrm{x}}^2-9\mathrm{x}+14}-\frac{4}{\mathrm{x}-7}$

Answer 1

Find the difference in the simplest form:

$\frac{7\mathrm{x}}{{\mathrm{x}}^2-9\mathrm{x}+14}-\frac{4}{\mathrm{x}-7}$

Use splitting the middle term to factorize ${\mathrm{x}}^2-9\mathrm{x}+14$

Find two numbers such that their sum is $-9$ and the product is $14$.

$$\begin{gathered} {\mathrm{x}}^2-7\mathrm{x}-2\mathrm{x}+14=\mathrm{x}(\mathrm{x}-7)-2(\mathrm{x}-7) \\ =(\mathrm{x}-7)(\mathrm{x}-2) \end{gathered}$$

Put them in the given expression:

$$\begin{gathered} \frac{7\mathrm{x}}{(\mathrm{x}-7)(\mathrm{x}-2)}-\frac{4}{\mathrm{x}-7}=\frac{7\mathrm{x}-4(\mathrm{x}-2)}{(\mathrm{x}-7)(\mathrm{x}-2)} \\ =\frac{7\mathrm{x}-4\mathrm{x}+8}{(\mathrm{x}-7)(\mathrm{x}-2)} \\ =\frac{3\mathrm{x}+8}{(\mathrm{x}-7)(\mathrm{x}-2)}\left[\mathrm{subtract}\mathrm{the}\mathrm{like}\mathrm{terms}\right] \end{gathered}$$

Hence, the difference in the simplest form is $\frac{\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{8}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{7}\mathbf{)}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{)}}$.