We want to simplify the expression:
[tex]\frac{6a^2-24a+24}{6a^2-24}[/tex]We could simplify this equation factoring its denominator and its numerator.
First, let's factor the numerator as follows:
[tex]6a^2-24a+24[/tex]Start multiplying and dividing the equation by 6 and then re-write it as:
[tex]\frac{6(6a^2-24a+24)}{6}=\frac{(6a)^2-24(6a)+144}{6}[/tex]Now, we're going to ask two numbers, whose sum is -24 and its multiplication is 144.
These numbers are -12 and -12. We can put these numbers in our previous equation like this:
[tex]\frac{(6a-12)(6a-12)}{6}[/tex]Now, we could apply common factor to this expression:
[tex]\frac{6(a-2)(6a-12)}{6}=(a-2)(6a-12)[/tex]And, we're going to simplify the denominator of the rational expression applying common factor too and then using the square difference expression like this:
[tex](6a^2-24)=6(a^2-4)=6(a+2)(a-2)[/tex]Finally, our rational expression can be simplified as:
[tex]\frac{6a^2-24a+24}{6a^2-24}=\frac{6(a-2)(a-2)}{6(a+2)(a-2)}=\frac{a-2}{a+2}[/tex]Therefore, the answers are:
- The numerator is a-2
- The denominator is a+2