Solve 2v/v+2+5/v+3

$$\frac { 2 v } { v + 2 } + \frac { 5 } { v + 3 }$$

Evaluate

$$\frac{2v^{2}+11v+10}{\left(v+2\right)\left(v+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of v+2 and v+3 is \left(v+2\right)\left(v+3\right). Multiply \frac{2v}{v+2} times \frac{v+3}{v+3}. Multiply \frac{5}{v+3} times \frac{v+2}{v+2}.

$$\frac{2v\left(v+3\right)}{\left(v+2\right)\left(v+3\right)}+\frac{5\left(v+2\right)}{\left(v+2\right)\left(v+3\right)}$$

Since \frac{2v\left(v+3\right)}{\left(v+2\right)\left(v+3\right)} and \frac{5\left(v+2\right)}{\left(v+2\right)\left(v+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{2v\left(v+3\right)+5\left(v+2\right)}{\left(v+2\right)\left(v+3\right)}$$

Do the multiplications in 2v\left(v+3\right)+5\left(v+2\right).

$$\frac{2v^{2}+6v+5v+10}{\left(v+2\right)\left(v+3\right)}$$

Combine like terms in 2v^{2}+6v+5v+10.

$$\frac{2v^{2}+11v+10}{\left(v+2\right)\left(v+3\right)}$$

Expand \left(v+2\right)\left(v+3\right).

$$\frac{2v^{2}+11v+10}{v^{2}+5v+6}$$

Differentiate w.r.t. v

$$\frac{16+4v-v^{2}}{v^{4}+10v^{3}+37v^{2}+60v+36}$$