Solve -7/v^2-4v-21+3/v^2-49

$$\frac { - 7 } { v ^ { 2 } - 4 v - 21 } + \frac { 3 } { v ^ { 2 } - 49 }$$

Evaluate

$$-\frac{4\left(v+10\right)}{\left(v+3\right)\left(v^{2}-49\right)}$$

Short Solution Steps

Factor v^{2}-4v-21. Factor v^{2}-49.

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

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

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

Since \frac{-7\left(v+7\right)}{\left(v-7\right)\left(v+3\right)\left(v+7\right)} and \frac{3\left(v+3\right)}{\left(v-7\right)\left(v+3\right)\left(v+7\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -7\left(v+7\right)+3\left(v+3\right).

$$\frac{-7v-49+3v+9}{\left(v-7\right)\left(v+3\right)\left(v+7\right)}$$

Combine like terms in -7v-49+3v+9.

$$\frac{-4v-40}{\left(v-7\right)\left(v+3\right)\left(v+7\right)}$$

Expand \left(v-7\right)\left(v+3\right)\left(v+7\right).

$$\frac{-4v-40}{v^{3}+3v^{2}-49v-147}$$

Differentiate w.r.t. v

$$\frac{4\left(2v^{3}+33v^{2}+60v-343\right)}{\left(\left(v+3\right)\left(v^{2}-49\right)\right)^{2}}$$