Solve x/x+2+4/x-5

$$\frac { x } { x + 2 } + \frac { 4 } { x - 5 }$$

Evaluate

$$\frac{x^{2}-x+8}{\left(x-5\right)\left(x+2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x-5 is \left(x-5\right)\left(x+2\right). Multiply \frac{x}{x+2} times \frac{x-5}{x-5}. Multiply \frac{4}{x-5} times \frac{x+2}{x+2}.

$$\frac{x\left(x-5\right)}{\left(x-5\right)\left(x+2\right)}+\frac{4\left(x+2\right)}{\left(x-5\right)\left(x+2\right)}$$

Since \frac{x\left(x-5\right)}{\left(x-5\right)\left(x+2\right)} and \frac{4\left(x+2\right)}{\left(x-5\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{x\left(x-5\right)+4\left(x+2\right)}{\left(x-5\right)\left(x+2\right)}$$

Do the multiplications in x\left(x-5\right)+4\left(x+2\right).

$$\frac{x^{2}-5x+4x+8}{\left(x-5\right)\left(x+2\right)}$$

Combine like terms in x^{2}-5x+4x+8.

$$\frac{x^{2}-x+8}{\left(x-5\right)\left(x+2\right)}$$

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

$$\frac{x^{2}-x+8}{x^{2}-3x-10}$$

Differentiate w.r.t. x

$$-\frac{2\left(\left(x+9\right)^{2}-98\right)}{\left(\left(x-5\right)\left(x+2\right)\right)^{2}}$$