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

1 Answer
Jun 7, 2015

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)