Solve 2x/x^2-9x+20+2x/x-4-x/x-5

$$\frac { 2 x } { x ^ { 2 } - 9 x + 20 } + \frac { 2 x } { x - 4 } - \frac { x } { x - 5 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-9x+20.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x-4\right) and x-4 is \left(x-5\right)\left(x-4\right). Multiply \frac{2x}{x-4} times \frac{x-5}{x-5}.

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

Since \frac{2x}{\left(x-5\right)\left(x-4\right)} and \frac{2x\left(x-5\right)}{\left(x-5\right)\left(x-4\right)} have the same denominator, add them by adding their numerators.

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

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

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

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

$$\frac{-8x+2x^{2}}{\left(x-5\right)\left(x-4\right)}-\frac{x}{x-5}$$

Factor the expressions that are not already factored in \frac{-8x+2x^{2}}{\left(x-5\right)\left(x-4\right)}.

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

Cancel out x-4 in both numerator and denominator.

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

Since \frac{2x}{x-5} and \frac{x}{x-5} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in 2x-x.

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

Factor

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