Solve x/x-2-8/x2-4

$$\frac{ x }{ x-2 } - \frac{ 8 }{ x2-4 }$$

Evaluate

$$\frac{xx_{2}-12x+16}{\left(x-2\right)\left(x_{2}-4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x_{2}-4 is \left(x-2\right)\left(x_{2}-4\right). Multiply \frac{x}{x-2} times \frac{x_{2}-4}{x_{2}-4}. Multiply \frac{8}{x_{2}-4} times \frac{x-2}{x-2}.

$$\frac{x\left(x_{2}-4\right)}{\left(x-2\right)\left(x_{2}-4\right)}-\frac{8\left(x-2\right)}{\left(x-2\right)\left(x_{2}-4\right)}$$

Since \frac{x\left(x_{2}-4\right)}{\left(x-2\right)\left(x_{2}-4\right)} and \frac{8\left(x-2\right)}{\left(x-2\right)\left(x_{2}-4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x_{2}-4\right)-8\left(x-2\right)}{\left(x-2\right)\left(x_{2}-4\right)}$$

Do the multiplications in x\left(x_{2}-4\right)-8\left(x-2\right).

$$\frac{xx_{2}-4x-8x+16}{\left(x-2\right)\left(x_{2}-4\right)}$$

Combine like terms in xx_{2}-4x-8x+16.

$$\frac{xx_{2}-12x+16}{\left(x-2\right)\left(x_{2}-4\right)}$$

Expand \left(x-2\right)\left(x_{2}-4\right).

$$\frac{xx_{2}-12x+16}{xx_{2}-4x-2x_{2}+8}$$

Differentiate w.r.t. x

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