Solve (x+5/x^2-25+1/x+5)(2x^2-13x+15/4x^2-6x)

$$( \frac { x + 5 } { x ^ { 2 } - 25 } + \frac { 1 } { x + 5 } ) ( \frac { 2 x ^ { 2 } - 13 x + 15 } { 4 x ^ { 2 } - 6 x } )$$

Evaluate

$$\frac{1}{x+5}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x+5}{x^{2}-25}.

$$\left(\frac{x+5}{\left(x-5\right)\left(x+5\right)}+\frac{1}{x+5}\right)\times \frac{2x^{2}-13x+15}{4x^{2}-6x}$$

Cancel out x+5 in both numerator and denominator.

$$\left(\frac{1}{x-5}+\frac{1}{x+5}\right)\times \frac{2x^{2}-13x+15}{4x^{2}-6x}$$

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

$$\left(\frac{x+5}{\left(x-5\right)\left(x+5\right)}+\frac{x-5}{\left(x-5\right)\left(x+5\right)}\right)\times \frac{2x^{2}-13x+15}{4x^{2}-6x}$$

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

$$\frac{x+5+x-5}{\left(x-5\right)\left(x+5\right)}\times \frac{2x^{2}-13x+15}{4x^{2}-6x}$$

Combine like terms in x+5+x-5.

$$\frac{2x}{\left(x-5\right)\left(x+5\right)}\times \frac{2x^{2}-13x+15}{4x^{2}-6x}$$

Factor the expressions that are not already factored in \frac{2x^{2}-13x+15}{4x^{2}-6x}.

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

Cancel out 2x-3 in both numerator and denominator.

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

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

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

Cancel out 2x\left(x-5\right) in both numerator and denominator.

$$\frac{1}{x+5}$$

Expand

$$\frac{1}{x+5}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x+5}{x^{2}-25}.

$$\left(\frac{x+5}{\left(x-5\right)\left(x+5\right)}+\frac{1}{x+5}\right)\times \frac{2x^{2}-13x+15}{4x^{2}-6x}$$

Cancel out x+5 in both numerator and denominator.

$$\left(\frac{1}{x-5}+\frac{1}{x+5}\right)\times \frac{2x^{2}-13x+15}{4x^{2}-6x}$$

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

$$\left(\frac{x+5}{\left(x-5\right)\left(x+5\right)}+\frac{x-5}{\left(x-5\right)\left(x+5\right)}\right)\times \frac{2x^{2}-13x+15}{4x^{2}-6x}$$

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

$$\frac{x+5+x-5}{\left(x-5\right)\left(x+5\right)}\times \frac{2x^{2}-13x+15}{4x^{2}-6x}$$

Combine like terms in x+5+x-5.

$$\frac{2x}{\left(x-5\right)\left(x+5\right)}\times \frac{2x^{2}-13x+15}{4x^{2}-6x}$$

Factor the expressions that are not already factored in \frac{2x^{2}-13x+15}{4x^{2}-6x}.

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

Cancel out 2x-3 in both numerator and denominator.

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

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

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

Cancel out 2x\left(x-5\right) in both numerator and denominator.

$$\frac{1}{x+5}$$