Question

Express the following difference in the simplest form $?$

$\frac{3}{x-4}-\frac{x-9}{x^2-16}$

Answer 1

Find the difference in the simplest form:

$\frac{3}{\mathrm{x}-4}-\frac{\mathrm{x}-9}{{\mathrm{x}}^2-16}$

Use the identity ${\mathrm{a}}^2-{\mathrm{b}}^2=(\mathrm{a}-\mathrm{b})(\mathrm{a}+\mathrm{b})$ to simplify ${\mathrm{x}}^2-16$:

$x^2-16=(x-4)(x+4)$

Put them in the given expression:

$\frac{3}{\mathrm{x}-4}-\frac{\mathrm{x}-9}{(\mathrm{x}-4)(\mathrm{x}+4)}$

$$\begin{gathered} =\frac{3(\mathrm{x}+4)-(\mathrm{x}-9)}{(\mathrm{x}-4)(\mathrm{x}+4)} \\ =\frac{3\mathrm{x}+12-\mathrm{x}+9}{(\mathrm{x}-4)(\mathrm{x}+4)} \\ =\frac{3\mathrm{x}-\mathrm{x}+12+9}{(\mathrm{x}-4)(\mathrm{x}+4)} \end{gathered}$$(simplify the like terms by grouping them)

$=\frac{2\mathrm{x}+21}{(\mathrm{x}+4)(\mathrm{x}-4)}$

Hence, the difference in the simplest form is $\frac{\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{21}}{\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{4}\mathbf{)}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{4}\mathbf{)}}$.