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

Jun 7, 2015

We can first factor the numerator :

$2 m - 6 = \textcolor{p u r p \le}{2 \left(m - 3\right)}$

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

• $- 2 a b = - 6 m$
Thus $2 a b = 6 m$ ( we multiply by $- 1$ on both side )
$a b = 3 m$ ( we divide by $2$ on both side )
$m \cdot b = 3 \cdot m$ ( since color(red)(a=m )
$\cancel{m} \cdot b = 3 \cdot \cancel{m}$
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)

$= \frac{\textcolor{p u r p \le}{2 \cancel{m - 3}}}{\left(\textcolor{red}{m} - \textcolor{b l u e}{3}\right) \cancel{\textcolor{red}{m} - \textcolor{b l u e}{3}}}$

$= \frac{2}{m - 3}$