Solve (2(1-x)/x^{2-4}+2/x-2-3/x^2+2x)*2x^3-9x^2+10x/12x^2-60x+75

$$( \frac { 2 ( 1 - x ) } { x ^ { 2 } - 4 } + \frac { 2 } { x - 2 } - \frac { 3 } { x ^ { 2 } + 2 x } ) \cdot \frac { 2 x ^ { 3 } - 9 x ^ { 2 } + 10 x } { 12 x ^ { 2 } - 60 x + 75 }$$

Evaluate

$$\frac{1}{2x-5}$$

Short Solution Steps

Factor x^{2}-4.

$$\left(\frac{2\left(1-x\right)}{\left(x-2\right)\left(x+2\right)}+\frac{2}{x-2}-\frac{3}{x^{2}+2x}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

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

$$\left(\frac{2\left(1-x\right)}{\left(x-2\right)\left(x+2\right)}+\frac{2\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{3}{x^{2}+2x}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

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

$$\left(\frac{2\left(1-x\right)+2\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{3}{x^{2}+2x}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Do the multiplications in 2\left(1-x\right)+2\left(x+2\right).

$$\left(\frac{2-2x+2x+4}{\left(x-2\right)\left(x+2\right)}-\frac{3}{x^{2}+2x}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Combine like terms in 2-2x+2x+4.

$$\left(\frac{6}{\left(x-2\right)\left(x+2\right)}-\frac{3}{x^{2}+2x}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Factor x^{2}+2x.

$$\left(\frac{6}{\left(x-2\right)\left(x+2\right)}-\frac{3}{x\left(x+2\right)}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

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

$$\left(\frac{6x}{x\left(x-2\right)\left(x+2\right)}-\frac{3\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Since \frac{6x}{x\left(x-2\right)\left(x+2\right)} and \frac{3\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6x-3\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)}\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Do the multiplications in 6x-3\left(x-2\right).

$$\frac{6x-3x+6}{x\left(x-2\right)\left(x+2\right)}\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Combine like terms in 6x-3x+6.

$$\frac{3x+6}{x\left(x-2\right)\left(x+2\right)}\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Factor the expressions that are not already factored in \frac{3x+6}{x\left(x-2\right)\left(x+2\right)}.

$$\frac{3\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{3}{x\left(x-2\right)}\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Factor the expressions that are not already factored in \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}.

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

Cancel out 2x-5 in both numerator and denominator.

$$\frac{3}{x\left(x-2\right)}\times \frac{x\left(x-2\right)}{3\left(2x-5\right)}$$

Multiply \frac{3}{x\left(x-2\right)} times \frac{x\left(x-2\right)}{3\left(2x-5\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3x\left(x-2\right)}{x\left(x-2\right)\times 3\left(2x-5\right)}$$

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

$$\frac{1}{2x-5}$$

Expand

$$\frac{1}{2\left(x-\frac{5}{2}\right)}$$

Short Solution Steps

Factor x^{2}-4.

$$\left(\frac{2\left(1-x\right)}{\left(x-2\right)\left(x+2\right)}+\frac{2}{x-2}-\frac{3}{x^{2}+2x}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

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

$$\left(\frac{2\left(1-x\right)}{\left(x-2\right)\left(x+2\right)}+\frac{2\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{3}{x^{2}+2x}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

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

$$\left(\frac{2\left(1-x\right)+2\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{3}{x^{2}+2x}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Do the multiplications in 2\left(1-x\right)+2\left(x+2\right).

$$\left(\frac{2-2x+2x+4}{\left(x-2\right)\left(x+2\right)}-\frac{3}{x^{2}+2x}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Combine like terms in 2-2x+2x+4.

$$\left(\frac{6}{\left(x-2\right)\left(x+2\right)}-\frac{3}{x^{2}+2x}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Factor x^{2}+2x.

$$\left(\frac{6}{\left(x-2\right)\left(x+2\right)}-\frac{3}{x\left(x+2\right)}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

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

$$\left(\frac{6x}{x\left(x-2\right)\left(x+2\right)}-\frac{3\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)}\right)\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Since \frac{6x}{x\left(x-2\right)\left(x+2\right)} and \frac{3\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6x-3\left(x-2\right)}{x\left(x-2\right)\left(x+2\right)}\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Do the multiplications in 6x-3\left(x-2\right).

$$\frac{6x-3x+6}{x\left(x-2\right)\left(x+2\right)}\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Combine like terms in 6x-3x+6.

$$\frac{3x+6}{x\left(x-2\right)\left(x+2\right)}\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Factor the expressions that are not already factored in \frac{3x+6}{x\left(x-2\right)\left(x+2\right)}.

$$\frac{3\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{3}{x\left(x-2\right)}\times \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}$$

Factor the expressions that are not already factored in \frac{2x^{3}-9x^{2}+10x}{12x^{2}-60x+75}.

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

Cancel out 2x-5 in both numerator and denominator.

$$\frac{3}{x\left(x-2\right)}\times \frac{x\left(x-2\right)}{3\left(2x-5\right)}$$

Multiply \frac{3}{x\left(x-2\right)} times \frac{x\left(x-2\right)}{3\left(2x-5\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3x\left(x-2\right)}{x\left(x-2\right)\times 3\left(2x-5\right)}$$

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

$$\frac{1}{2x-5}$$