The answer is:
[tex]\frac{f(x)}{g(x)}=\frac{x+3}{x+4}[/tex]
To solve the problem, we need to factorize the quadratic functions in order to be able to simplify the expression.
We can factorize quadratic functions in the following way:
[tex]a^{2}-b^{2} =(a-b)(a+b)[/tex]
Also, we can factorize/simplify quadratic expressions in the following way, if we have the following quadratic expression:
[tex]ax^{2}+bx+c[/tex]
We can factorize it by finding two numbers which its products give as result "c" (j) and its algebraic sum gives as result "b" (k), and then, rewrite the expression in the following way:
[tex](x+j)(x+k)[/tex]
Where,
x, is the variable.
j, is the first obtained value.
k, is the second obtained value.
We are given the functions:
[tex]f(x)=x^{2} -9\\g(x)=x^{2} -7x+12[/tex]
Then, factoring we have:
First expression,
[tex]f(x)=x^{2} -9=(x+3)(x-3)[/tex]
Second expression,
[tex]\g(x)=x^{2} -7x+12[/tex]
We need to find two number which product gives as result 12 and their algebraic sum gives as result -7. Those numbers are -4 and -3.
[tex]-4*-3=12\\-4-3=-7[/tex]
Now, rewriting the expression we have:
[tex]\g(x)=x^{2} -7x+12=(x-4)(x-3)[/tex]
So, solving we have:
[tex](f/g)(x)=\frac{f(x)}{g(x)}=\frac{(x+3)(x-3)}{(x+4)(x-3)}=\frac{x+3}{x+4}[/tex]
Have a nice day!
