Solve z-13/z^2-z-6-2/3-z

$$\frac { z - 13 } { z ^ { 2 } - z - 6 } - \frac { 2 } { 3 - z }$$

Evaluate

$$\frac{3}{z+2}$$

Short Solution Steps

Factor z^{2}-z-6.

$$\frac{z-13}{\left(z-3\right)\left(z+2\right)}-\frac{2}{3-z}$$

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

$$\frac{z-13}{\left(z-3\right)\left(z+2\right)}-\frac{2\left(-1\right)\left(z+2\right)}{\left(z-3\right)\left(z+2\right)}$$

Since \frac{z-13}{\left(z-3\right)\left(z+2\right)} and \frac{2\left(-1\right)\left(z+2\right)}{\left(z-3\right)\left(z+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{z-13-2\left(-1\right)\left(z+2\right)}{\left(z-3\right)\left(z+2\right)}$$

Do the multiplications in z-13-2\left(-1\right)\left(z+2\right).

$$\frac{z-13+2z+4}{\left(z-3\right)\left(z+2\right)}$$

Combine like terms in z-13+2z+4.

$$\frac{3z-9}{\left(z-3\right)\left(z+2\right)}$$

Factor the expressions that are not already factored in \frac{3z-9}{\left(z-3\right)\left(z+2\right)}.

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

Cancel out z-3 in both numerator and denominator.

$$\frac{3}{z+2}$$

Expand

$$\frac{3}{z+2}$$

Short Solution Steps

Factor z^{2}-z-6.

$$\frac{z-13}{\left(z-3\right)\left(z+2\right)}-\frac{2}{3-z}$$

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

$$\frac{z-13}{\left(z-3\right)\left(z+2\right)}-\frac{2\left(-1\right)\left(z+2\right)}{\left(z-3\right)\left(z+2\right)}$$

Since \frac{z-13}{\left(z-3\right)\left(z+2\right)} and \frac{2\left(-1\right)\left(z+2\right)}{\left(z-3\right)\left(z+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{z-13-2\left(-1\right)\left(z+2\right)}{\left(z-3\right)\left(z+2\right)}$$

Do the multiplications in z-13-2\left(-1\right)\left(z+2\right).

$$\frac{z-13+2z+4}{\left(z-3\right)\left(z+2\right)}$$

Combine like terms in z-13+2z+4.

$$\frac{3z-9}{\left(z-3\right)\left(z+2\right)}$$

Factor the expressions that are not already factored in \frac{3z-9}{\left(z-3\right)\left(z+2\right)}.

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

Cancel out z-3 in both numerator and denominator.

$$\frac{3}{z+2}$$