Solve 3/y^2-y-6/frac{2{y-3}+4/y+2}

$$\frac { \frac { 3 } { y ^ { 2 } - y - 6 } } { \frac { 2 } { y - 3 } + \frac { 4 } { y + 2 } }$$

Evaluate

$$\frac{3}{2\left(3y-4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y-3 and y+2 is \left(y-3\right)\left(y+2\right). Multiply \frac{2}{y-3} times \frac{y+2}{y+2}. Multiply \frac{4}{y+2} times \frac{y-3}{y-3}.

$$\frac{\frac{3}{y^{2}-y-6}}{\frac{2\left(y+2\right)}{\left(y-3\right)\left(y+2\right)}+\frac{4\left(y-3\right)}{\left(y-3\right)\left(y+2\right)}}$$

Since \frac{2\left(y+2\right)}{\left(y-3\right)\left(y+2\right)} and \frac{4\left(y-3\right)}{\left(y-3\right)\left(y+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3}{y^{2}-y-6}}{\frac{2\left(y+2\right)+4\left(y-3\right)}{\left(y-3\right)\left(y+2\right)}}$$

Do the multiplications in 2\left(y+2\right)+4\left(y-3\right).

$$\frac{\frac{3}{y^{2}-y-6}}{\frac{2y+4+4y-12}{\left(y-3\right)\left(y+2\right)}}$$

Combine like terms in 2y+4+4y-12.

$$\frac{\frac{3}{y^{2}-y-6}}{\frac{6y-8}{\left(y-3\right)\left(y+2\right)}}$$

Divide \frac{3}{y^{2}-y-6} by \frac{6y-8}{\left(y-3\right)\left(y+2\right)} by multiplying \frac{3}{y^{2}-y-6} by the reciprocal of \frac{6y-8}{\left(y-3\right)\left(y+2\right)}.

$$\frac{3\left(y-3\right)\left(y+2\right)}{\left(y^{2}-y-6\right)\left(6y-8\right)}$$

Factor the expressions that are not already factored.

$$\frac{3\left(y-3\right)\left(y+2\right)}{2\left(y-3\right)\left(3y-4\right)\left(y+2\right)}$$

Cancel out \left(y-3\right)\left(y+2\right) in both numerator and denominator.

$$\frac{3}{2\left(3y-4\right)}$$

Expand the expression.

$$\frac{3}{6y-8}$$

Differentiate w.r.t. y

$$-\frac{9}{2\left(3y-4\right)^{2}}$$