Question

How do you simplify `(r+6)/(4^2-r-6)` and state the restrictions?

Answer 1

I assume your question is `(r + 6)/(r^2 - r - 6)`

Let's Factorise the `color(red)(DENOMINATOR)` first.

The Denominator is `color(red)(r^2 - r - 6)`

We can use Splitting the Middle Term technique to factorise this.

It is in the form `ax^2 + bx + c` where `a=1, b=-1, c= -6`

To split the middle term, we need to think of two numbers `N_1 and N_2` such that:
`N_1*N_2 = a*c and N_1+N_2 = b`
`N_1*N_2 = (1)*(-6) and N_1+N_2 = -1`
`N_1*N_2 = -6 and N_1+N_2 = -1`

After Trial and Error, we get `N_1 = 2 and N_2 = -3`
`(2)*(-3) = -6` and `(2) + (-3) = -1`

So we can write the denominator as
`color(red)(r^2 +2r -3r - 6)`
`= r*(r+2) - 3*(r+2)`
`= (r+2)*(r-3)`

The Denominator can be written as `color(red)((r-2)*(r+3))`

The expression we have been given is
`(r + 6)/(r^2 - r - 6)`

After the denominator was factorised, the Expression can now be written as :

`((r+6))/((r+2)*(r-3))`

`(r + 6)/(r^2 - r - 6)` = `((r+6))/((r+2)*(r-3))`