Solve x+5/2x-2-1/x^2-3x+2-1/x^2-4x

$$\frac { x + 5 } { 2 x - 2 } - \frac { 1 } { x ^ { 2 } - 3 x + 2 } - \frac { 1 } { x ^ { 2 } - 4 x }$$

Evaluate

$$\frac{x^{4}-x^{3}-26x^{2}+54x-4}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}$$

Short Solution Steps

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

$$\frac{x+5}{2\left(x-1\right)}-\frac{1}{\left(x-2\right)\left(x-1\right)}-\frac{1}{x^{2}-4x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-1\right) and \left(x-2\right)\left(x-1\right) is 2\left(x-2\right)\left(x-1\right). Multiply \frac{x+5}{2\left(x-1\right)} times \frac{x-2}{x-2}. Multiply \frac{1}{\left(x-2\right)\left(x-1\right)} times \frac{2}{2}.

$$\frac{\left(x+5\right)\left(x-2\right)}{2\left(x-2\right)\left(x-1\right)}-\frac{2}{2\left(x-2\right)\left(x-1\right)}-\frac{1}{x^{2}-4x}$$

Since \frac{\left(x+5\right)\left(x-2\right)}{2\left(x-2\right)\left(x-1\right)} and \frac{2}{2\left(x-2\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+5\right)\left(x-2\right)-2}{2\left(x-2\right)\left(x-1\right)}-\frac{1}{x^{2}-4x}$$

Do the multiplications in \left(x+5\right)\left(x-2\right)-2.

$$\frac{x^{2}-2x+5x-10-2}{2\left(x-2\right)\left(x-1\right)}-\frac{1}{x^{2}-4x}$$

Combine like terms in x^{2}-2x+5x-10-2.

$$\frac{x^{2}+3x-12}{2\left(x-2\right)\left(x-1\right)}-\frac{1}{x^{2}-4x}$$

Factor x^{2}-4x.

$$\frac{x^{2}+3x-12}{2\left(x-2\right)\left(x-1\right)}-\frac{1}{x\left(x-4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-2\right)\left(x-1\right) and x\left(x-4\right) is 2x\left(x-4\right)\left(x-2\right)\left(x-1\right). Multiply \frac{x^{2}+3x-12}{2\left(x-2\right)\left(x-1\right)} times \frac{x\left(x-4\right)}{x\left(x-4\right)}. Multiply \frac{1}{x\left(x-4\right)} times \frac{2\left(x-2\right)\left(x-1\right)}{2\left(x-2\right)\left(x-1\right)}.

$$\frac{\left(x^{2}+3x-12\right)x\left(x-4\right)}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}-\frac{2\left(x-2\right)\left(x-1\right)}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}$$

Since \frac{\left(x^{2}+3x-12\right)x\left(x-4\right)}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)} and \frac{2\left(x-2\right)\left(x-1\right)}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}+3x-12\right)x\left(x-4\right)-2\left(x-2\right)\left(x-1\right)}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}$$

Do the multiplications in \left(x^{2}+3x-12\right)x\left(x-4\right)-2\left(x-2\right)\left(x-1\right).

$$\frac{x^{4}-4x^{3}+3x^{3}-12x^{2}-12x^{2}+48x-2x^{2}+2x+4x-4}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}$$

Combine like terms in x^{4}-4x^{3}+3x^{3}-12x^{2}-12x^{2}+48x-2x^{2}+2x+4x-4.

$$\frac{x^{4}-x^{3}-26x^{2}+54x-4}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}$$

Expand 2x\left(x-4\right)\left(x-2\right)\left(x-1\right).

$$\frac{x^{4}-x^{3}-26x^{2}+54x-4}{2x^{4}-14x^{3}+28x^{2}-16x}$$

Expand

$$\frac{x^{4}-x^{3}-26x^{2}+54x-4}{2\left(x-2\right)\left(x-1\right)\left(x^{2}-4x\right)}$$

Short Solution Steps

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

$$\frac{x+5}{2\left(x-1\right)}-\frac{1}{\left(x-2\right)\left(x-1\right)}-\frac{1}{x^{2}-4x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-1\right) and \left(x-2\right)\left(x-1\right) is 2\left(x-2\right)\left(x-1\right). Multiply \frac{x+5}{2\left(x-1\right)} times \frac{x-2}{x-2}. Multiply \frac{1}{\left(x-2\right)\left(x-1\right)} times \frac{2}{2}.

$$\frac{\left(x+5\right)\left(x-2\right)}{2\left(x-2\right)\left(x-1\right)}-\frac{2}{2\left(x-2\right)\left(x-1\right)}-\frac{1}{x^{2}-4x}$$

Since \frac{\left(x+5\right)\left(x-2\right)}{2\left(x-2\right)\left(x-1\right)} and \frac{2}{2\left(x-2\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+5\right)\left(x-2\right)-2}{2\left(x-2\right)\left(x-1\right)}-\frac{1}{x^{2}-4x}$$

Do the multiplications in \left(x+5\right)\left(x-2\right)-2.

$$\frac{x^{2}-2x+5x-10-2}{2\left(x-2\right)\left(x-1\right)}-\frac{1}{x^{2}-4x}$$

Combine like terms in x^{2}-2x+5x-10-2.

$$\frac{x^{2}+3x-12}{2\left(x-2\right)\left(x-1\right)}-\frac{1}{x^{2}-4x}$$

Factor x^{2}-4x.

$$\frac{x^{2}+3x-12}{2\left(x-2\right)\left(x-1\right)}-\frac{1}{x\left(x-4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-2\right)\left(x-1\right) and x\left(x-4\right) is 2x\left(x-4\right)\left(x-2\right)\left(x-1\right). Multiply \frac{x^{2}+3x-12}{2\left(x-2\right)\left(x-1\right)} times \frac{x\left(x-4\right)}{x\left(x-4\right)}. Multiply \frac{1}{x\left(x-4\right)} times \frac{2\left(x-2\right)\left(x-1\right)}{2\left(x-2\right)\left(x-1\right)}.

$$\frac{\left(x^{2}+3x-12\right)x\left(x-4\right)}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}-\frac{2\left(x-2\right)\left(x-1\right)}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}$$

Since \frac{\left(x^{2}+3x-12\right)x\left(x-4\right)}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)} and \frac{2\left(x-2\right)\left(x-1\right)}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{2}+3x-12\right)x\left(x-4\right)-2\left(x-2\right)\left(x-1\right)}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}$$

Do the multiplications in \left(x^{2}+3x-12\right)x\left(x-4\right)-2\left(x-2\right)\left(x-1\right).

$$\frac{x^{4}-4x^{3}+3x^{3}-12x^{2}-12x^{2}+48x-2x^{2}+2x+4x-4}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}$$

Combine like terms in x^{4}-4x^{3}+3x^{3}-12x^{2}-12x^{2}+48x-2x^{2}+2x+4x-4.

$$\frac{x^{4}-x^{3}-26x^{2}+54x-4}{2x\left(x-4\right)\left(x-2\right)\left(x-1\right)}$$

Expand 2x\left(x-4\right)\left(x-2\right)\left(x-1\right).

$$\frac{x^{4}-x^{3}-26x^{2}+54x-4}{2x^{4}-14x^{3}+28x^{2}-16x}$$