Solve 2x^2-x/x^2-x-30-2/x+5+x/6-x

$$\frac { 2 x ^ { 2 } - x } { x ^ { 2 } - x - 30 } - \frac { 2 } { x + 5 } + \frac { x } { 6 - x }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-x-30.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-6\right)\left(x+5\right) and x+5 is \left(x-6\right)\left(x+5\right). Multiply \frac{2}{x+5} times \frac{x-6}{x-6}.

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

Since \frac{2x^{2}-x}{\left(x-6\right)\left(x+5\right)} and \frac{2\left(x-6\right)}{\left(x-6\right)\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{2x^{2}-x-2x+12}{\left(x-6\right)\left(x+5\right)}+\frac{x}{6-x}$$

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

$$\frac{2x^{2}-3x+12}{\left(x-6\right)\left(x+5\right)}+\frac{x}{6-x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-6\right)\left(x+5\right) and 6-x is \left(x-6\right)\left(x+5\right). Multiply \frac{x}{6-x} times \frac{-\left(x+5\right)}{-\left(x+5\right)}.

$$\frac{2x^{2}-3x+12}{\left(x-6\right)\left(x+5\right)}+\frac{x\left(-1\right)\left(x+5\right)}{\left(x-6\right)\left(x+5\right)}$$

Since \frac{2x^{2}-3x+12}{\left(x-6\right)\left(x+5\right)} and \frac{x\left(-1\right)\left(x+5\right)}{\left(x-6\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x^{2}-3x+12+x\left(-1\right)\left(x+5\right)}{\left(x-6\right)\left(x+5\right)}$$

Do the multiplications in 2x^{2}-3x+12+x\left(-1\right)\left(x+5\right).

$$\frac{2x^{2}-3x+12-x^{2}-5x}{\left(x-6\right)\left(x+5\right)}$$

Combine like terms in 2x^{2}-3x+12-x^{2}-5x.

$$\frac{x^{2}-8x+12}{\left(x-6\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{x^{2}-8x+12}{\left(x-6\right)\left(x+5\right)}.

$$\frac{\left(x-6\right)\left(x-2\right)}{\left(x-6\right)\left(x+5\right)}$$

Cancel out x-6 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor x^{2}-x-30.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-6\right)\left(x+5\right) and x+5 is \left(x-6\right)\left(x+5\right). Multiply \frac{2}{x+5} times \frac{x-6}{x-6}.

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

Since \frac{2x^{2}-x}{\left(x-6\right)\left(x+5\right)} and \frac{2\left(x-6\right)}{\left(x-6\right)\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{2x^{2}-x-2x+12}{\left(x-6\right)\left(x+5\right)}+\frac{x}{6-x}$$

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

$$\frac{2x^{2}-3x+12}{\left(x-6\right)\left(x+5\right)}+\frac{x}{6-x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-6\right)\left(x+5\right) and 6-x is \left(x-6\right)\left(x+5\right). Multiply \frac{x}{6-x} times \frac{-\left(x+5\right)}{-\left(x+5\right)}.

$$\frac{2x^{2}-3x+12}{\left(x-6\right)\left(x+5\right)}+\frac{x\left(-1\right)\left(x+5\right)}{\left(x-6\right)\left(x+5\right)}$$

Since \frac{2x^{2}-3x+12}{\left(x-6\right)\left(x+5\right)} and \frac{x\left(-1\right)\left(x+5\right)}{\left(x-6\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x^{2}-3x+12+x\left(-1\right)\left(x+5\right)}{\left(x-6\right)\left(x+5\right)}$$

Do the multiplications in 2x^{2}-3x+12+x\left(-1\right)\left(x+5\right).

$$\frac{2x^{2}-3x+12-x^{2}-5x}{\left(x-6\right)\left(x+5\right)}$$

Combine like terms in 2x^{2}-3x+12-x^{2}-5x.

$$\frac{x^{2}-8x+12}{\left(x-6\right)\left(x+5\right)}$$

Factor the expressions that are not already factored in \frac{x^{2}-8x+12}{\left(x-6\right)\left(x+5\right)}.

$$\frac{\left(x-6\right)\left(x-2\right)}{\left(x-6\right)\left(x+5\right)}$$

Cancel out x-6 in both numerator and denominator.

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