Solve 2/x-6-48/x^2-6

$$\frac { 2 } { x - 6 } - \frac { 48 } { x ^ { 2 } - 6 }$$

Evaluate

$$\frac{2\left(x^{2}-24x+138\right)}{\left(x-6\right)\left(x^{2}-6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-6 and x^{2}-6 is \left(x-6\right)\left(x^{2}-6\right). Multiply \frac{2}{x-6} times \frac{x^{2}-6}{x^{2}-6}. Multiply \frac{48}{x^{2}-6} times \frac{x-6}{x-6}.

$$\frac{2\left(x^{2}-6\right)}{\left(x-6\right)\left(x^{2}-6\right)}-\frac{48\left(x-6\right)}{\left(x-6\right)\left(x^{2}-6\right)}$$

Since \frac{2\left(x^{2}-6\right)}{\left(x-6\right)\left(x^{2}-6\right)} and \frac{48\left(x-6\right)}{\left(x-6\right)\left(x^{2}-6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(x^{2}-6\right)-48\left(x-6\right)}{\left(x-6\right)\left(x^{2}-6\right)}$$

Do the multiplications in 2\left(x^{2}-6\right)-48\left(x-6\right).

$$\frac{2x^{2}-12-48x+288}{\left(x-6\right)\left(x^{2}-6\right)}$$

Combine like terms in 2x^{2}-12-48x+288.

$$\frac{2x^{2}+276-48x}{\left(x-6\right)\left(x^{2}-6\right)}$$

Expand \left(x-6\right)\left(x^{2}-6\right).

$$\frac{2x^{2}+276-48x}{x^{3}-6x^{2}-6x+36}$$

Factor

$$\frac{2\left(x-\left(12-\sqrt{6}\right)\right)\left(x-\left(\sqrt{6}+12\right)\right)}{\left(x-6\right)\left(x^{2}-6\right)}$$