How do you simplify `(x^2-12x+36)/(4x-24)`?

Look for potential of parts of the numerator also occurring in the denominator so that you can cancel out.

Consider the numerator: `x^2-12x+36`

To get anything at all that starts to looks like the denominator we have to factorise. Lets try it!

Try: `(x+6)(x-6)`

That does not work. Why is that?
Look at the constants: `(-6) times (+6)` gives `-36`. That is wrong so we need to change something.

Try: `(x-6)(x-6) = x^2 -12x +36` Now we have found the factors.

Substitute this into our original equation giving:

`((x-6)(x-6))/(4x-24)`

We will have found what we want if we factor out 4 from the denominator giving:

`((x-6)(x-6))/(4(x-6))`

Write as:

`((x-6))/4 times ((x-6))/((x-6))`

But:

`((x-6))/((x-6)) = 1`

giving:

`((x-6))/4`

The numerator `(x^2-12x+36)` is in the form `a^2-2ab+b^2`, where `a=x and b=6`.

Rewrite the numerator.

`((x^2)-2(x)(6)+(6^2))/(4x-24)`

Apply the square of a difference `(a-b)^2=a^2-2ab+b^2`.

`(x-6)^2/(4x-24)`

Factor `4` out of the denominator.

`(x-6)^2/(4(x-6))`

Rewrite the numerator.

`((x-6)(x-6))/(4(x-6))`

Cancel `(x-6)` from the numerator and denominator.

`((cancel(x-6))(x-6))/(4cancel((x-6))``=`

`(x-6)/4`