Solve x/x^2-4+1/x-3

$$\frac { x } { x ^ { 2 } - 4 } + \frac { 1 } { x - 3 }$$

Evaluate

$$\frac{2x^{2}-3x-4}{\left(x-3\right)\left(x^{2}-4\right)}$$

Short Solution Steps

Factor x^{2}-4.

$$\frac{x}{\left(x-2\right)\left(x+2\right)}+\frac{1}{x-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+2\right) and x-3 is \left(x-3\right)\left(x-2\right)\left(x+2\right). Multiply \frac{x}{\left(x-2\right)\left(x+2\right)} times \frac{x-3}{x-3}. Multiply \frac{1}{x-3} times \frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}.

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

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

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

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

$$\frac{x^{2}-3x+x^{2}+2x-2x-4}{\left(x-3\right)\left(x-2\right)\left(x+2\right)}$$

Combine like terms in x^{2}-3x+x^{2}+2x-2x-4.

$$\frac{2x^{2}-3x-4}{\left(x-3\right)\left(x-2\right)\left(x+2\right)}$$

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

$$\frac{2x^{2}-3x-4}{x^{3}-3x^{2}-4x+12}$$

Differentiate w.r.t. x

$$\frac{-2x^{4}+6x^{3}-5x^{2}+24x-52}{\left(\left(x-3\right)\left(x^{2}-4\right)\right)^{2}}$$