Question

Express the difference below in the simplest form $?$

$\frac{\mathrm{d}+2}{4\mathrm{d}-1}-\frac{7}{\mathrm{d}+5}$

Answer 1

Find the difference in the simplest form:

$\frac{\mathrm{d}+2}{4\mathrm{d}-1}-\frac{7}{\mathrm{d}+5}$ (simplify by taking the least common denominator)

$\frac{(\mathrm{d}+2)(\mathrm{d}+5)-7(4\mathrm{d}-1)}{(4\mathrm{d}-1)(\mathrm{d}+5)}$ (simplify by distributing terms)

$$\begin{gathered} =\frac{\mathrm{d}\left(\right(\mathrm{d}+5)+2(\mathrm{d}+5)-28\mathrm{d}+7}{(4\mathrm{d}-1)(\mathrm{d}+5)} \\ =\frac{{\mathrm{d}}^2+5\mathrm{d}+2\mathrm{d}+10-28\mathrm{d}+7}{(4\mathrm{d}-1)(\mathrm{d}+5)} \\ =\frac{{\mathrm{d}}^2+(5\mathrm{d}+2\mathrm{d}-28\mathrm{d})+(10+7)}{(4\mathrm{d}-1)(\mathrm{d}+5)} \end{gathered}$$ (group the like terms and combine them together)

$=\frac{{\mathrm{d}}^2-21\mathrm{d}+17}{(4\mathrm{d}-1)(\mathrm{d}+5)}$

Hence, the difference in the simplest form is $\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{-}\mathbf{21}\mathbf{d}\mathbf{+}\mathbf{17}}{\mathbf{(}\mathbf{4}\mathbf{d}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{d}\mathbf{+}\mathbf{5}\mathbf{)}}$