Solve 5/3p+1-6p/p-5

$$\frac { 5 } { 3 p + 1 } - \frac { 6 p } { p - 5 }$$

Evaluate

$$-\frac{18p^{2}+p+25}{\left(p-5\right)\left(3p+1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3p+1 and p-5 is \left(p-5\right)\left(3p+1\right). Multiply \frac{5}{3p+1} times \frac{p-5}{p-5}. Multiply \frac{6p}{p-5} times \frac{3p+1}{3p+1}.

$$\frac{5\left(p-5\right)}{\left(p-5\right)\left(3p+1\right)}-\frac{6p\left(3p+1\right)}{\left(p-5\right)\left(3p+1\right)}$$

Since \frac{5\left(p-5\right)}{\left(p-5\right)\left(3p+1\right)} and \frac{6p\left(3p+1\right)}{\left(p-5\right)\left(3p+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(p-5\right)-6p\left(3p+1\right)}{\left(p-5\right)\left(3p+1\right)}$$

Do the multiplications in 5\left(p-5\right)-6p\left(3p+1\right).

$$\frac{5p-25-18p^{2}-6p}{\left(p-5\right)\left(3p+1\right)}$$

Combine like terms in 5p-25-18p^{2}-6p.

$$\frac{-p-25-18p^{2}}{\left(p-5\right)\left(3p+1\right)}$$

Expand \left(p-5\right)\left(3p+1\right).

$$\frac{-p-25-18p^{2}}{3p^{2}-14p-5}$$

Differentiate w.r.t. p

$$\frac{15\left(17p^{2}+22p-23\right)}{9p^{4}-84p^{3}+166p^{2}+140p+25}$$