# How do you solve 2/(y+2)+3/y=-y/(y+2)?

Jan 5, 2017

$x = 1.5 \mathmr{and} - 6$

#### Explanation:

$\frac{2}{y + 2} + \frac{3}{y} = - \frac{y}{y + 2}$

find LCM of $y$ and $y + 2$

factors of $y$: $y , 1$

factors of $y + 2$: $y + 2 , 1$

multiply together $y , 1 \mathmr{and} \left(y + 2\right)$:

$y \cdot 1 \cdot \left(y + 2\right) = y \left(y + 2\right)$

multiply denominators to get $y \left(y + 2\right)$:

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

multiply everything by $y \left(y + 2\right)$

hence, $2 y + 3 \left(y + 2\right) = - \left({y}^{2}\right)$

multiply out brackets:

$2 y + 3 y + 6 = - \left({y}^{2}\right)$

$5 y + 6 = - \left({y}^{2}\right)$

multiply by $- 1$:

$- \left(5 y + 6\right) = {y}^{2}$

$6 - 5 y = {y}^{2}$

add $5 y$:

${y}^{2} + 5 y = 6$

complete the square:

halve the $y$ term $\left(2.5\right)$.

square it $\left(6.25\right)$ and add to both sides.

$5 x + 6.25 + {x}^{2} = 6 + 6.25$

${x}^{2} + 5 x + 6.25 = 12.25$

factorise:
$\left(x + 2.5\right) \left(x + 2.5\right) = 12.25$

calculate the square root of the right side $\left(3.5 \mathmr{and} - 3.5\right)$

$\therefore x + 2.5 = 3.5 \mathmr{and} - 3.5$

subtract $2.5$:

$x = 1.5 \mathmr{and} - 6$