Question

Express the following difference in the simplest form $?$

$\frac{{\mathrm{d}}^2-11}{{\mathrm{d}}^2-7\mathrm{d}+12}-\frac{\mathrm{d}+1}{\mathrm{d}-4}$

Answer 1

Find the difference in the simplest form:

$\frac{{\mathrm{d}}^2-11}{{\mathrm{d}}^2-7\mathrm{d}+12}-\frac{\mathrm{d}+1}{\mathrm{d}-4}$

Use splitting the middle term to factorize $({\mathrm{d}}^2-7\mathrm{d}+12)$:

(Take two numbers such that their sum is $-7$ and their product is $12$)

${\mathrm{d}}^2-3\mathrm{d}-4\mathrm{d}+12$ (take the common factor)

$$\begin{gathered} =\mathrm{d}(\mathrm{d}-3)-4(\mathrm{d}-3) \\ =(\mathrm{d}-4)(\mathrm{d}-3) \end{gathered}$$

Substitute in the given expression:

$\frac{{\mathrm{d}}^2-11}{(\mathrm{d}-4)(\mathrm{d}-3)}-\frac{\mathrm{d}+1}{\mathrm{d}-4}$ (simplify by taking the least common denominator)

$=\frac{{\mathrm{d}}^2-11-(\mathrm{d}-3)(\mathrm{d}+1)}{(\mathrm{d}-4)(\mathrm{d}-3)}$ (simplify by distribution)

$$\begin{gathered} =\frac{{\mathrm{d}}^2-11-\left[\mathrm{d}\right(\mathrm{d}+1)-3(\mathrm{d}+1\left)\right]}{(\mathrm{d}-4)(\mathrm{d}-3)} \\ =\frac{{\mathrm{d}}^2-11-[{\mathrm{d}}^2+\mathrm{d}-3\mathrm{d}-3]}{(\mathrm{d}-4)(\mathrm{d}-3)} \\ =\frac{{\mathrm{d}}^2-11-{\mathrm{d}}^2-\mathrm{d}+3\mathrm{d}+3}{(\mathrm{d}-4)(\mathrm{d}-3)} \end{gathered}$$

$=\frac{({\mathrm{d}}^2-{\mathrm{d}}^2)+(3\mathrm{d}-\mathrm{d})+(-11+3)}{(\mathrm{d}-4)(\mathrm{d}-3)}$ (group the like terms together and combine them)

$=\frac{2\mathrm{d}-8}{(\mathrm{d}-4)(\mathrm{d}-3)}$ (Now, take the common term in the numerator)

$=\frac{2(\mathrm{d}-4)}{(\mathrm{d}-4)(\mathrm{d}-3)}$ (cancel the like term)

$=\frac{2}{\mathrm{d}-3}$

Hence, the difference in the lowest form is $\frac{\mathbf{2}}{\mathbf{d}\mathbf{-}\mathbf{3}}$.