Solve (x-2/3x+6+1/x^2-4+x-6/6-3x)*9x^2-36/32=

$$( \frac { x - 2 } { 3 x + 6 } + \frac { 1 } { x ^ { 2 } - 4 } + \frac { x - 6 } { 6 - 3 x } ) \cdot \frac { 9 x ^ { 2 } - 36 } { 32 } =$$

Evaluate

$$\frac{57}{32}=1.78125$$

Short Solution Steps

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

$$\left(\frac{x-2}{3\left(x+2\right)}+\frac{1}{\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-36}{32}$$

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}.

$$\left(\frac{\left(x-2\right)\left(x-2\right)}{3\left(x-2\right)\left(x+2\right)}+\frac{3}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-36}{32}$$

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.

$$\left(\frac{\left(x-2\right)\left(x-2\right)+3}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-36}{32}$$

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

$$\left(\frac{x^{2}-2x-2x+4+3}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-36}{32}$$

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

$$\left(\frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{6-3x}\right)\times \frac{9x^{2}-36}{32}$$

Factor 6-3x.

$$\left(\frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)}+\frac{x-6}{3\left(-x+2\right)}\right)\times \frac{9x^{2}-36}{32}$$

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)}.

$$\left(\frac{x^{2}-4x+7}{3\left(x-2\right)\left(x+2\right)}+\frac{\left(x-6\right)\left(-1\right)\left(x+2\right)}{3\left(x-2\right)\left(x+2\right)}\right)\times \frac{9x^{2}-36}{32}$$

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.

$$\frac{x^{2}-4x+7+\left(x-6\right)\left(-1\right)\left(x+2\right)}{3\left(x-2\right)\left(x+2\right)}\times \frac{9x^{2}-36}{32}$$

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

$$\frac{x^{2}-4x+7-x^{2}-2x+6x+12}{3\left(x-2\right)\left(x+2\right)}\times \frac{9x^{2}-36}{32}$$

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

$$\frac{19}{3\left(x-2\right)\left(x+2\right)}\times \frac{9x^{2}-36}{32}$$

Multiply \frac{19}{3\left(x-2\right)\left(x+2\right)} times \frac{9x^{2}-36}{32} by multiplying numerator times numerator and denominator times denominator.

$$\frac{19\left(9x^{2}-36\right)}{3\left(x-2\right)\left(x+2\right)\times 32}$$

Multiply 3 and 32 to get 96.

$$\frac{19\left(9x^{2}-36\right)}{96\left(x-2\right)\left(x+2\right)}$$

Factor the expressions that are not already factored.

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

Cancel out 3\left(x-2\right)\left(x+2\right) in both numerator and denominator.

$$\frac{3\times 19}{32}$$

Multiply 3 and 19 to get 57.

$$\frac{57}{32}$$

Factor

$$\frac{3 \cdot 19}{2 ^ {5}} = 1\frac{25}{32} = 1.78125$$