Question

How do you simplify `(2m-6)/(m^2-6m+9)`?

Answer 1

We can first factor the numerator :

`2m-6=color(purple)(2(m-3))`

We also know that `(color(red)a-color(blue)b)²=color(red)(a²)-2color(red)acolor(blue)b+color(blue)(b²)`

We can thus factor `m²-6m+9` :

• `a²=m²`
  So `color(red)(a=m`

• `-2ab=-6m`
  Thus `2ab=6m` ( we multiply by `-1` on both side )
  `ab=3m` ( we divide by `2` on both side )
  `m*b=3*m` ( since `color(red)(a=m` )
  `cancelm*b=3*cancelm`
  `color(blue)(b=3`

• `9=b²`
  So `color(blue)(b=3` : What we found, perfect.

Thus :
`m²-6m+9=(color(red)a-color(blue)b)²=(color(red)m-color(blue)3)²`

And thus :

`(2m-6)/(m²-6m+9)=color(purple)(2(m-3))/((color(red)m-color(blue)3)²)`

`=color(purple)(2cancel(m-3))/((color(red)m-color(blue)3)cancel(color(red)m-color(blue)3))`

`=2/(m-3)`