Solve left(x-2/3x+6+1/x^2-4+x-6/6-3xright)9x^2-x/32

$$\left( \frac{ x-2 }{ 3x+6 } + \frac{ 1 }{ { x }^{ 2 } -4 } + \frac{ x-6 }{ 6-3x } \right) \frac{ 9 { x }^{ 2 } -x }{ 32 }$$

Evaluate

$$\frac{19x\left(9x-1\right)}{96\left(x^{2}-4\right)}$$

Short Solution Steps

Factor 3x+6. Factor x^{2}-4.

$$\left(\frac{x-2}{3\left(x+2\right)}+\frac{1}{\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-x}{32}$$

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

$$\left(\frac{\left(x-2\right)\left(x-2\right)}{3\left(x-2\right)\left(x+2\right)}+\frac{3}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-x}{32}$$

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

$$\left(\frac{\left(x-2\right)\left(x-2\right)+3}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-x}{32}$$

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

$$\left(\frac{x^{2}-2x-2x+4+3}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-x}{32}$$

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

$$\left(\frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-x}{32}$$

Factor 6-3x.

$$\left(\frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{3\left(-x+2\right)}\right)\times \frac{9x^{2}-x}{32}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x-2\right)\left(x+2\right) and 3\left(-x+2\right) is 3\left(x-2\right)\left(x+2\right). Multiply \frac{x-6}{3\left(-x+2\right)} times \frac{-\left(x+2\right)}{-\left(x+2\right)}.

$$\left(\frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)}+\frac{\left(x-6\right)\left(-1\right)\left(x+2\right)}{3\left(x-2\right)\left(x+2\right)}\right)\times \frac{9x^{2}-x}{32}$$

Since \frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)} and \frac{\left(x-6\right)\left(-1\right)\left(x+2\right)}{3\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}-4x+7+\left(x-6\right)\left(-1\right)\left(x+2\right)}{3\left(x-2\right)\left(x+2\right)}\times \frac{9x^{2}-x}{32}$$

Do the multiplications in x^{2}-4x+7+\left(x-6\right)\left(-1\right)\left(x+2\right).

$$\frac{x^{2}-4x+7-x^{2}-2x+6x+12}{3\left(x-2\right)\left(x+2\right)}\times \frac{9x^{2}-x}{32}$$

Combine like terms in x^{2}-4x+7-x^{2}-2x+6x+12.

$$\frac{19}{3\left(x-2\right)\left(x+2\right)}\times \frac{9x^{2}-x}{32}$$

Multiply \frac{19}{3\left(x-2\right)\left(x+2\right)} times \frac{9x^{2}-x}{32} by multiplying numerator times numerator and denominator times denominator.

$$\frac{19\left(9x^{2}-x\right)}{3\left(x-2\right)\left(x+2\right)\times 32}$$

Multiply 3 and 32 to get 96.

$$\frac{19\left(9x^{2}-x\right)}{96\left(x-2\right)\left(x+2\right)}$$

Use the distributive property to multiply 19 by 9x^{2}-x.

$$\frac{171x^{2}-19x}{96\left(x-2\right)\left(x+2\right)}$$

Use the distributive property to multiply 96 by x-2.

$$\frac{171x^{2}-19x}{\left(96x-192\right)\left(x+2\right)}$$

Use the distributive property to multiply 96x-192 by x+2 and combine like terms.

$$\frac{171x^{2}-19x}{96x^{2}-384}$$

Expand

$$\frac{19\left(9x^{2}-x\right)}{96\left(x^{2}-4\right)}$$

Short Solution Steps

Factor 3x+6. Factor x^{2}-4.

$$\left(\frac{x-2}{3\left(x+2\right)}+\frac{1}{\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-x}{32}$$

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

$$\left(\frac{\left(x-2\right)\left(x-2\right)}{3\left(x-2\right)\left(x+2\right)}+\frac{3}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-x}{32}$$

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

$$\left(\frac{\left(x-2\right)\left(x-2\right)+3}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-x}{32}$$

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

$$\left(\frac{x^{2}-2x-2x+4+3}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-x}{32}$$

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

$$\left(\frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-x}{32}$$

Factor 6-3x.

$$\left(\frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{3\left(-x+2\right)}\right)\times \frac{9x^{2}-x}{32}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3\left(x-2\right)\left(x+2\right) and 3\left(-x+2\right) is 3\left(x-2\right)\left(x+2\right). Multiply \frac{x-6}{3\left(-x+2\right)} times \frac{-\left(x+2\right)}{-\left(x+2\right)}.

$$\left(\frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)}+\frac{\left(x-6\right)\left(-1\right)\left(x+2\right)}{3\left(x-2\right)\left(x+2\right)}\right)\times \frac{9x^{2}-x}{32}$$

Since \frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)} and \frac{\left(x-6\right)\left(-1\right)\left(x+2\right)}{3\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}-4x+7+\left(x-6\right)\left(-1\right)\left(x+2\right)}{3\left(x-2\right)\left(x+2\right)}\times \frac{9x^{2}-x}{32}$$

Do the multiplications in x^{2}-4x+7+\left(x-6\right)\left(-1\right)\left(x+2\right).

$$\frac{x^{2}-4x+7-x^{2}-2x+6x+12}{3\left(x-2\right)\left(x+2\right)}\times \frac{9x^{2}-x}{32}$$

Combine like terms in x^{2}-4x+7-x^{2}-2x+6x+12.

$$\frac{19}{3\left(x-2\right)\left(x+2\right)}\times \frac{9x^{2}-x}{32}$$

Multiply \frac{19}{3\left(x-2\right)\left(x+2\right)} times \frac{9x^{2}-x}{32} by multiplying numerator times numerator and denominator times denominator.

$$\frac{19\left(9x^{2}-x\right)}{3\left(x-2\right)\left(x+2\right)\times 32}$$

Multiply 3 and 32 to get 96.

$$\frac{19\left(9x^{2}-x\right)}{96\left(x-2\right)\left(x+2\right)}$$

Use the distributive property to multiply 19 by 9x^{2}-x.

$$\frac{171x^{2}-19x}{96\left(x-2\right)\left(x+2\right)}$$

Use the distributive property to multiply 96 by x-2.

$$\frac{171x^{2}-19x}{\left(96x-192\right)\left(x+2\right)}$$

Use the distributive property to multiply 96x-192 by x+2 and combine like terms.

$$\frac{171x^{2}-19x}{96x^{2}-384}$$