Solve 6x/x^2+x-12-x/x^2-16

$$\frac { 6 x } { x ^ { 2 } + x - 12 } - \frac { x } { x ^ { 2 } - 16 }$$

Evaluate

$$\frac{x\left(5x-21\right)}{\left(x-3\right)\left(x^{2}-16\right)}$$

Short Solution Steps

Factor x^{2}+x-12. Factor x^{2}-16.

$$\frac{6x}{\left(x-3\right)\left(x+4\right)}-\frac{x}{\left(x-4\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+4\right) and \left(x-4\right)\left(x+4\right) is \left(x-4\right)\left(x-3\right)\left(x+4\right). Multiply \frac{6x}{\left(x-3\right)\left(x+4\right)} times \frac{x-4}{x-4}. Multiply \frac{x}{\left(x-4\right)\left(x+4\right)} times \frac{x-3}{x-3}.

$$\frac{6x\left(x-4\right)}{\left(x-4\right)\left(x-3\right)\left(x+4\right)}-\frac{x\left(x-3\right)}{\left(x-4\right)\left(x-3\right)\left(x+4\right)}$$

Since \frac{6x\left(x-4\right)}{\left(x-4\right)\left(x-3\right)\left(x+4\right)} and \frac{x\left(x-3\right)}{\left(x-4\right)\left(x-3\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6x\left(x-4\right)-x\left(x-3\right)}{\left(x-4\right)\left(x-3\right)\left(x+4\right)}$$

Do the multiplications in 6x\left(x-4\right)-x\left(x-3\right).

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

Combine like terms in 6x^{2}-24x-x^{2}+3x.

$$\frac{5x^{2}-21x}{\left(x-4\right)\left(x-3\right)\left(x+4\right)}$$

Expand \left(x-4\right)\left(x-3\right)\left(x+4\right).

$$\frac{5x^{2}-21x}{x^{3}-3x^{2}-16x+48}$$

Factor

$$\frac{x\left(5x-21\right)}{\left(x-3\right)\left(x^{2}-16\right)}$$