Solve 3/w^2-4-2w/w^2-7w+10+6/w^2-3w-10

$$\frac { 3 } { w ^ { 2 } - 4 } - \frac { 2 w } { w ^ { 2 } - 7 w + 10 } + \frac { 6 } { w ^ { 2 } - 3 w - 10 }$$

Evaluate

$$\frac{-2w^{2}+5w-27}{\left(w-5\right)\left(w^{2}-4\right)}$$

Short Solution Steps

Factor w^{2}-4. Factor w^{2}-7w+10.

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

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

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

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

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

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

$$\frac{3w-15-2w^{2}-4w}{\left(w-5\right)\left(w-2\right)\left(w+2\right)}+\frac{6}{w^{2}-3w-10}$$

Combine like terms in 3w-15-2w^{2}-4w.

$$\frac{-w-15-2w^{2}}{\left(w-5\right)\left(w-2\right)\left(w+2\right)}+\frac{6}{w^{2}-3w-10}$$

Factor w^{2}-3w-10.

$$\frac{-w-15-2w^{2}}{\left(w-5\right)\left(w-2\right)\left(w+2\right)}+\frac{6}{\left(w-5\right)\left(w+2\right)}$$

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

$$\frac{-w-15-2w^{2}}{\left(w-5\right)\left(w-2\right)\left(w+2\right)}+\frac{6\left(w-2\right)}{\left(w-5\right)\left(w-2\right)\left(w+2\right)}$$

Since \frac{-w-15-2w^{2}}{\left(w-5\right)\left(w-2\right)\left(w+2\right)} and \frac{6\left(w-2\right)}{\left(w-5\right)\left(w-2\right)\left(w+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{-w-15-2w^{2}+6\left(w-2\right)}{\left(w-5\right)\left(w-2\right)\left(w+2\right)}$$

Do the multiplications in -w-15-2w^{2}+6\left(w-2\right).

$$\frac{-w-15-2w^{2}+6w-12}{\left(w-5\right)\left(w-2\right)\left(w+2\right)}$$

Combine like terms in -w-15-2w^{2}+6w-12.

$$\frac{5w-27-2w^{2}}{\left(w-5\right)\left(w-2\right)\left(w+2\right)}$$

Expand \left(w-5\right)\left(w-2\right)\left(w+2\right).

$$\frac{5w-27-2w^{2}}{w^{3}-5w^{2}-4w+20}$$

Differentiate w.r.t. w

$$\frac{2\left(w^{4}-5w^{3}+57w^{2}-175w-4\right)}{\left(\left(w-5\right)\left(w^{2}-4\right)\right)^{2}}$$