Solve 2/p-q-3/2p+q

$$\frac { 2 } { p - q } - \frac { 3 } { 2 p + q }$$

Evaluate

$$\frac{p+5q}{\left(p-q\right)\left(2p+q\right)}$$

Short Solution Steps

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

$$\frac{2\left(2p+q\right)}{\left(p-q\right)\left(2p+q\right)}-\frac{3\left(p-q\right)}{\left(p-q\right)\left(2p+q\right)}$$

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

$$\frac{2\left(2p+q\right)-3\left(p-q\right)}{\left(p-q\right)\left(2p+q\right)}$$

Do the multiplications in 2\left(2p+q\right)-3\left(p-q\right).

$$\frac{4p+2q-3p+3q}{\left(p-q\right)\left(2p+q\right)}$$

Combine like terms in 4p+2q-3p+3q.

$$\frac{p+5q}{\left(p-q\right)\left(2p+q\right)}$$

Expand \left(p-q\right)\left(2p+q\right).

$$\frac{p+5q}{2p^{2}-pq-q^{2}}$$

Differentiate w.r.t. p

$$\frac{2\left(2q^{2}-10pq-p^{2}\right)}{\left(\left(p-q\right)\left(2p+q\right)\right)^{2}}$$