Solve z+2/z-2-z-5/z+6-4/z^2+4z-12

$$\frac { z + 2 } { z - 2 } - \frac { z - 5 } { z + 6 } - \frac { 4 } { z ^ { 2 } + 4 z - 12 }$$

Evaluate

$$\frac{15z-2}{\left(z-2\right)\left(z+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of z-2 and z+6 is \left(z-2\right)\left(z+6\right). Multiply \frac{z+2}{z-2} times \frac{z+6}{z+6}. Multiply \frac{z-5}{z+6} times \frac{z-2}{z-2}.

$$\frac{\left(z+2\right)\left(z+6\right)}{\left(z-2\right)\left(z+6\right)}-\frac{\left(z-5\right)\left(z-2\right)}{\left(z-2\right)\left(z+6\right)}-\frac{4}{z^{2}+4z-12}$$

Since \frac{\left(z+2\right)\left(z+6\right)}{\left(z-2\right)\left(z+6\right)} and \frac{\left(z-5\right)\left(z-2\right)}{\left(z-2\right)\left(z+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(z+2\right)\left(z+6\right)-\left(z-5\right)\left(z-2\right)}{\left(z-2\right)\left(z+6\right)}-\frac{4}{z^{2}+4z-12}$$

Do the multiplications in \left(z+2\right)\left(z+6\right)-\left(z-5\right)\left(z-2\right).

$$\frac{z^{2}+6z+2z+12-z^{2}+2z+5z-10}{\left(z-2\right)\left(z+6\right)}-\frac{4}{z^{2}+4z-12}$$

Combine like terms in z^{2}+6z+2z+12-z^{2}+2z+5z-10.

$$\frac{15z+2}{\left(z-2\right)\left(z+6\right)}-\frac{4}{z^{2}+4z-12}$$

Factor z^{2}+4z-12.

$$\frac{15z+2}{\left(z-2\right)\left(z+6\right)}-\frac{4}{\left(z-2\right)\left(z+6\right)}$$

Since \frac{15z+2}{\left(z-2\right)\left(z+6\right)} and \frac{4}{\left(z-2\right)\left(z+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{15z+2-4}{\left(z-2\right)\left(z+6\right)}$$

Combine like terms in 15z+2-4.

$$\frac{15z-2}{\left(z-2\right)\left(z+6\right)}$$

Expand \left(z-2\right)\left(z+6\right).

$$\frac{15z-2}{z^{2}+4z-12}$$

Expand

$$\frac{15z-2}{\left(z-2\right)\left(z+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of z-2 and z+6 is \left(z-2\right)\left(z+6\right). Multiply \frac{z+2}{z-2} times \frac{z+6}{z+6}. Multiply \frac{z-5}{z+6} times \frac{z-2}{z-2}.

$$\frac{\left(z+2\right)\left(z+6\right)}{\left(z-2\right)\left(z+6\right)}-\frac{\left(z-5\right)\left(z-2\right)}{\left(z-2\right)\left(z+6\right)}-\frac{4}{z^{2}+4z-12}$$

Since \frac{\left(z+2\right)\left(z+6\right)}{\left(z-2\right)\left(z+6\right)} and \frac{\left(z-5\right)\left(z-2\right)}{\left(z-2\right)\left(z+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(z+2\right)\left(z+6\right)-\left(z-5\right)\left(z-2\right)}{\left(z-2\right)\left(z+6\right)}-\frac{4}{z^{2}+4z-12}$$

Do the multiplications in \left(z+2\right)\left(z+6\right)-\left(z-5\right)\left(z-2\right).

$$\frac{z^{2}+6z+2z+12-z^{2}+2z+5z-10}{\left(z-2\right)\left(z+6\right)}-\frac{4}{z^{2}+4z-12}$$

Combine like terms in z^{2}+6z+2z+12-z^{2}+2z+5z-10.

$$\frac{15z+2}{\left(z-2\right)\left(z+6\right)}-\frac{4}{z^{2}+4z-12}$$

Factor z^{2}+4z-12.

$$\frac{15z+2}{\left(z-2\right)\left(z+6\right)}-\frac{4}{\left(z-2\right)\left(z+6\right)}$$

Since \frac{15z+2}{\left(z-2\right)\left(z+6\right)} and \frac{4}{\left(z-2\right)\left(z+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{15z+2-4}{\left(z-2\right)\left(z+6\right)}$$

Combine like terms in 15z+2-4.

$$\frac{15z-2}{\left(z-2\right)\left(z+6\right)}$$

Expand \left(z-2\right)\left(z+6\right).

$$\frac{15z-2}{z^{2}+4z-12}$$