Given the functions
[tex]\begin{gathered} f(x)=x^2-6x+9 \\ g(x)=x-3 \end{gathered}[/tex]
You have to find the quotient between both functions, that is (f/g)(x)
[tex](\frac{f}{g})(x)=\frac{x^2-6x+9}{x-3}[/tex]
To solve this division, the first step is to factor the numerator.
To factor the quadratic function, you have to find a value or values whose sum is -6 and their product is 9.
The number that fulfills both characteristics is -3
The factor of f(x) is (x-3) and its factorized form is:
[tex]\begin{gathered} f(x)=x^2-6x+9 \\ f(x)=(x-3)^2 \end{gathered}[/tex]
You can rewrite the quotient as follows:
[tex]\begin{gathered} (\frac{f}{g})(x)=\frac{x^2-6x+9}{x-3} \\ (\frac{f}{g})(x)=\frac{(x-3)^2}{x-3} \end{gathered}[/tex]
The next step is to simplify the expression:
[tex]\begin{gathered} (\frac{f}{g})(x)=\frac{(x-3)^{\bcancel{2 }}}{\bcancel{x-3 }} \\ (\frac{f}{g})(x)=x-3 \end{gathered}[/tex]
The result is (f/g)(x)=x-3
