Question

Given that f(x) = x^2 - 8x + 12 and g(x) = x – 6, find (f ÷ g)(x) and express the result in standard form.

Answer 1

$\begin{gathered} f(x)=x^2-8x+12 \\ g(x)=x-6_{} \end{gathered}$

f(x) can be factorized with help of the quadratic formula, as follows:

$\begin{gathered} x_{1,2}=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x_{1,2}=\frac{-(-8)\pm\sqrt[]{(-8)^2-4\cdot1\cdot12}}{2\cdot1} \\ x_{1,2}=\frac{8\pm\sqrt[]{64-48}}{2} \\ x_{1,2}=\frac{8\pm4}{2} \\ x_1=\frac{8+4}{2}=6 \\ x_2=\frac{8-4}{2}=2 \end{gathered}$

Then, f(x) can be expressed as: (x - 6)(x - 2). Replacing this into (f ÷ g)(x), we get:

$\frac{f\mleft(x\mright)}{g(x)}=\frac{x^2-8x+12}{x-6}=\frac{(x-6)(x-2)}{x-6}=x-2$