Solve 6v-1/v^2-5v-14-v-3/v^2-4

$$\frac { 6 v - 1 } { v ^ { 2 } - 5 v - 14 } - \frac { v - 3 } { v ^ { 2 } - 4 }$$

Evaluate

$$\frac{5v^{2}-3v-19}{\left(v-7\right)\left(v^{2}-4\right)}$$

Short Solution Steps

Factor v^{2}-5v-14. Factor v^{2}-4.

$$\frac{6v-1}{\left(v-7\right)\left(v+2\right)}-\frac{v-3}{\left(v-2\right)\left(v+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(v-7\right)\left(v+2\right) and \left(v-2\right)\left(v+2\right) is \left(v-7\right)\left(v-2\right)\left(v+2\right). Multiply \frac{6v-1}{\left(v-7\right)\left(v+2\right)} times \frac{v-2}{v-2}. Multiply \frac{v-3}{\left(v-2\right)\left(v+2\right)} times \frac{v-7}{v-7}.

$$\frac{\left(6v-1\right)\left(v-2\right)}{\left(v-7\right)\left(v-2\right)\left(v+2\right)}-\frac{\left(v-3\right)\left(v-7\right)}{\left(v-7\right)\left(v-2\right)\left(v+2\right)}$$

Since \frac{\left(6v-1\right)\left(v-2\right)}{\left(v-7\right)\left(v-2\right)\left(v+2\right)} and \frac{\left(v-3\right)\left(v-7\right)}{\left(v-7\right)\left(v-2\right)\left(v+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(6v-1\right)\left(v-2\right)-\left(v-3\right)\left(v-7\right)}{\left(v-7\right)\left(v-2\right)\left(v+2\right)}$$

Do the multiplications in \left(6v-1\right)\left(v-2\right)-\left(v-3\right)\left(v-7\right).

$$\frac{6v^{2}-12v-v+2-v^{2}+7v+3v-21}{\left(v-7\right)\left(v-2\right)\left(v+2\right)}$$

Combine like terms in 6v^{2}-12v-v+2-v^{2}+7v+3v-21.

$$\frac{5v^{2}-3v-19}{\left(v-7\right)\left(v-2\right)\left(v+2\right)}$$

Expand \left(v-7\right)\left(v-2\right)\left(v+2\right).

$$\frac{5v^{2}-3v-19}{v^{3}-7v^{2}-4v+28}$$

Expand

$$\frac{5v^{2}-3v-19}{\left(v-7\right)\left(v^{2}-4\right)}$$

Short Solution Steps

Factor v^{2}-5v-14. Factor v^{2}-4.

$$\frac{6v-1}{\left(v-7\right)\left(v+2\right)}-\frac{v-3}{\left(v-2\right)\left(v+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(v-7\right)\left(v+2\right) and \left(v-2\right)\left(v+2\right) is \left(v-7\right)\left(v-2\right)\left(v+2\right). Multiply \frac{6v-1}{\left(v-7\right)\left(v+2\right)} times \frac{v-2}{v-2}. Multiply \frac{v-3}{\left(v-2\right)\left(v+2\right)} times \frac{v-7}{v-7}.

$$\frac{\left(6v-1\right)\left(v-2\right)}{\left(v-7\right)\left(v-2\right)\left(v+2\right)}-\frac{\left(v-3\right)\left(v-7\right)}{\left(v-7\right)\left(v-2\right)\left(v+2\right)}$$

Since \frac{\left(6v-1\right)\left(v-2\right)}{\left(v-7\right)\left(v-2\right)\left(v+2\right)} and \frac{\left(v-3\right)\left(v-7\right)}{\left(v-7\right)\left(v-2\right)\left(v+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(6v-1\right)\left(v-2\right)-\left(v-3\right)\left(v-7\right)}{\left(v-7\right)\left(v-2\right)\left(v+2\right)}$$

Do the multiplications in \left(6v-1\right)\left(v-2\right)-\left(v-3\right)\left(v-7\right).

$$\frac{6v^{2}-12v-v+2-v^{2}+7v+3v-21}{\left(v-7\right)\left(v-2\right)\left(v+2\right)}$$

Combine like terms in 6v^{2}-12v-v+2-v^{2}+7v+3v-21.

$$\frac{5v^{2}-3v-19}{\left(v-7\right)\left(v-2\right)\left(v+2\right)}$$

Expand \left(v-7\right)\left(v-2\right)\left(v+2\right).

$$\frac{5v^{2}-3v-19}{v^{3}-7v^{2}-4v+28}$$