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

1 Answer
Tessalsifi
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)#

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