Solve 5v-1/v^2-5v-36-v-10/v^2-16

$$\frac { 5 v - 1 } { v ^ { 2 } - 5 v - 36 } - \frac { v - 10 } { v ^ { 2 } - 16 }$$

Evaluate

$$\frac{2\left(2v^{2}-v-43\right)}{\left(v-9\right)\left(v^{2}-16\right)}$$

Short Solution Steps

Factor v^{2}-5v-36. Factor v^{2}-16.

$$\frac{5v-1}{\left(v-9\right)\left(v+4\right)}-\frac{v-10}{\left(v-4\right)\left(v+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(v-9\right)\left(v+4\right) and \left(v-4\right)\left(v+4\right) is \left(v-9\right)\left(v-4\right)\left(v+4\right). Multiply \frac{5v-1}{\left(v-9\right)\left(v+4\right)} times \frac{v-4}{v-4}. Multiply \frac{v-10}{\left(v-4\right)\left(v+4\right)} times \frac{v-9}{v-9}.

$$\frac{\left(5v-1\right)\left(v-4\right)}{\left(v-9\right)\left(v-4\right)\left(v+4\right)}-\frac{\left(v-10\right)\left(v-9\right)}{\left(v-9\right)\left(v-4\right)\left(v+4\right)}$$

Since \frac{\left(5v-1\right)\left(v-4\right)}{\left(v-9\right)\left(v-4\right)\left(v+4\right)} and \frac{\left(v-10\right)\left(v-9\right)}{\left(v-9\right)\left(v-4\right)\left(v+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(5v-1\right)\left(v-4\right)-\left(v-10\right)\left(v-9\right)}{\left(v-9\right)\left(v-4\right)\left(v+4\right)}$$

Do the multiplications in \left(5v-1\right)\left(v-4\right)-\left(v-10\right)\left(v-9\right).

$$\frac{5v^{2}-20v-v+4-v^{2}+9v+10v-90}{\left(v-9\right)\left(v-4\right)\left(v+4\right)}$$

Combine like terms in 5v^{2}-20v-v+4-v^{2}+9v+10v-90.

$$\frac{4v^{2}-2v-86}{\left(v-9\right)\left(v-4\right)\left(v+4\right)}$$

Expand \left(v-9\right)\left(v-4\right)\left(v+4\right).

$$\frac{4v^{2}-2v-86}{v^{3}-9v^{2}-16v+144}$$

Expand

$$\frac{2\left(2v^{2}-v-43\right)}{\left(v-9\right)\left(v^{2}-16\right)}$$

Short Solution Steps

Factor v^{2}-5v-36. Factor v^{2}-16.

$$\frac{5v-1}{\left(v-9\right)\left(v+4\right)}-\frac{v-10}{\left(v-4\right)\left(v+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(v-9\right)\left(v+4\right) and \left(v-4\right)\left(v+4\right) is \left(v-9\right)\left(v-4\right)\left(v+4\right). Multiply \frac{5v-1}{\left(v-9\right)\left(v+4\right)} times \frac{v-4}{v-4}. Multiply \frac{v-10}{\left(v-4\right)\left(v+4\right)} times \frac{v-9}{v-9}.

$$\frac{\left(5v-1\right)\left(v-4\right)}{\left(v-9\right)\left(v-4\right)\left(v+4\right)}-\frac{\left(v-10\right)\left(v-9\right)}{\left(v-9\right)\left(v-4\right)\left(v+4\right)}$$

Since \frac{\left(5v-1\right)\left(v-4\right)}{\left(v-9\right)\left(v-4\right)\left(v+4\right)} and \frac{\left(v-10\right)\left(v-9\right)}{\left(v-9\right)\left(v-4\right)\left(v+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(5v-1\right)\left(v-4\right)-\left(v-10\right)\left(v-9\right)}{\left(v-9\right)\left(v-4\right)\left(v+4\right)}$$

Do the multiplications in \left(5v-1\right)\left(v-4\right)-\left(v-10\right)\left(v-9\right).

$$\frac{5v^{2}-20v-v+4-v^{2}+9v+10v-90}{\left(v-9\right)\left(v-4\right)\left(v+4\right)}$$

Combine like terms in 5v^{2}-20v-v+4-v^{2}+9v+10v-90.

$$\frac{4v^{2}-2v-86}{\left(v-9\right)\left(v-4\right)\left(v+4\right)}$$

Expand \left(v-9\right)\left(v-4\right)\left(v+4\right).

$$\frac{4v^{2}-2v-86}{v^{3}-9v^{2}-16v+144}$$