# How do you simplify (2y-6)/(3y^2-y^3)?

Dec 22, 2016

$\textcolor{g r e e n}{- \frac{2}{y} ^ 2}$

#### Explanation:

Note that $3 {y}^{2} - {y}^{3} = - {y}^{3} + 3 {y}^{2}$

$\textcolor{w h i t e}{\text{XXX}} = \left(- {y}^{2}\right) \left(y - 3\right)$

$\textcolor{w h i t e}{\text{XXX}} = \left(- {y}^{2}\right) \left(\frac{\left(2 y - 6\right)}{2}\right)$

$\textcolor{w h i t e}{\text{XXX}} = \left(- \frac{{y}^{2}}{2}\right) \left(2 y - 6\right)$

Therefore
$\textcolor{w h i t e}{\text{XXX}} \frac{2 y - 6}{- {y}^{3} + 3 {y}^{2}} = \frac{\cancel{2 y - 6}}{- {y}^{2} / 2 \left(\cancel{2 y - 6}\right)} = - \frac{2}{y} ^ 2$

Dec 22, 2016

$= - \frac{2}{y} ^ 2$

#### Explanation:

$\frac{2 y - 6}{3 {y}^{2} - {y}^{3}} \text{ }$ factorise as far as possible

$= \frac{2 \left(y - 3\right)}{{y}^{2} \left(3 - y\right)}$

Notice that the brackets are the same except for the signs.
Do a switch round:

$\textcolor{red}{\left(3 - y\right) = - \left(- 3 + y\right) = - \left(y - 3\right)}$

=(2(y-3))/(y^2color(red)((3-y))

=(2(y-3))/(-y^2color(red)((y-3))

=(2cancel((y-3)))/(-y^2cancel((y-3))

$= - \frac{2}{y} ^ 2$