Solve 5/x^2-2x-15-6/x^2+x-6

$$\frac { 5 } { x ^ { 2 } - 2 x - 15 } - \frac { 6 } { x ^ { 2 } + x - 6 }$$

Evaluate

$$-\frac{x-20}{\left(x-5\right)\left(x-2\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}-2x-15. Factor x^{2}+x-6.

$$\frac{5}{\left(x-5\right)\left(x+3\right)}-\frac{6}{\left(x-2\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x+3\right) and \left(x-2\right)\left(x+3\right) is \left(x-5\right)\left(x-2\right)\left(x+3\right). Multiply \frac{5}{\left(x-5\right)\left(x+3\right)} times \frac{x-2}{x-2}. Multiply \frac{6}{\left(x-2\right)\left(x+3\right)} times \frac{x-5}{x-5}.

$$\frac{5\left(x-2\right)}{\left(x-5\right)\left(x-2\right)\left(x+3\right)}-\frac{6\left(x-5\right)}{\left(x-5\right)\left(x-2\right)\left(x+3\right)}$$

Since \frac{5\left(x-2\right)}{\left(x-5\right)\left(x-2\right)\left(x+3\right)} and \frac{6\left(x-5\right)}{\left(x-5\right)\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\left(x-2\right)-6\left(x-5\right)}{\left(x-5\right)\left(x-2\right)\left(x+3\right)}$$

Do the multiplications in 5\left(x-2\right)-6\left(x-5\right).

$$\frac{5x-10-6x+30}{\left(x-5\right)\left(x-2\right)\left(x+3\right)}$$

Combine like terms in 5x-10-6x+30.

$$\frac{-x+20}{\left(x-5\right)\left(x-2\right)\left(x+3\right)}$$

Expand \left(x-5\right)\left(x-2\right)\left(x+3\right).

$$\frac{-x+20}{x^{3}-4x^{2}-11x+30}$$

Differentiate w.r.t. x

$$\frac{2\left(x^{3}-32x^{2}+80x+95\right)}{\left(\left(x-5\right)\left(x-2\right)\left(x+3\right)\right)^{2}}$$