Solve 2x+4/x^2-8x+15-x-5/x^2-x-6-x+3/x^2-3x-10

$$\frac { 2 x + 4 } { x ^ { 2 } - 8 x + 15 } - \frac { x - 5 } { x ^ { 2 } - x - 6 } - \frac { x + 3 } { x ^ { 2 } - 3 x - 10 }$$

Evaluate

$$\frac{2\left(9x-4\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}$$

Short Solution Steps

Factor x^{2}-8x+15. Factor x^{2}-x-6.

$$\frac{2x+4}{\left(x-5\right)\left(x-3\right)}-\frac{x-5}{\left(x-3\right)\left(x+2\right)}-\frac{x+3}{x^{2}-3x-10}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x-3\right) and \left(x-3\right)\left(x+2\right) is \left(x-5\right)\left(x-3\right)\left(x+2\right). Multiply \frac{2x+4}{\left(x-5\right)\left(x-3\right)} times \frac{x+2}{x+2}. Multiply \frac{x-5}{\left(x-3\right)\left(x+2\right)} times \frac{x-5}{x-5}.

$$\frac{\left(2x+4\right)\left(x+2\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{\left(x-5\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{x+3}{x^{2}-3x-10}$$

Since \frac{\left(2x+4\right)\left(x+2\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)} and \frac{\left(x-5\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x+4\right)\left(x+2\right)-\left(x-5\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{x+3}{x^{2}-3x-10}$$

Do the multiplications in \left(2x+4\right)\left(x+2\right)-\left(x-5\right)\left(x-5\right).

$$\frac{2x^{2}+4x+4x+8-x^{2}+5x+5x-25}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{x+3}{x^{2}-3x-10}$$

Combine like terms in 2x^{2}+4x+4x+8-x^{2}+5x+5x-25.

$$\frac{x^{2}+18x-17}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{x+3}{x^{2}-3x-10}$$

Factor x^{2}-3x-10.

$$\frac{x^{2}+18x-17}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{x+3}{\left(x-5\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x-3\right)\left(x+2\right) and \left(x-5\right)\left(x+2\right) is \left(x-5\right)\left(x-3\right)\left(x+2\right). Multiply \frac{x+3}{\left(x-5\right)\left(x+2\right)} times \frac{x-3}{x-3}.

$$\frac{x^{2}+18x-17}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{\left(x+3\right)\left(x-3\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}$$

Since \frac{x^{2}+18x-17}{\left(x-5\right)\left(x-3\right)\left(x+2\right)} and \frac{\left(x+3\right)\left(x-3\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+18x-17-\left(x+3\right)\left(x-3\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}$$

Do the multiplications in x^{2}+18x-17-\left(x+3\right)\left(x-3\right).

$$\frac{x^{2}+18x-17-x^{2}+3x-3x+9}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}$$

Combine like terms in x^{2}+18x-17-x^{2}+3x-3x+9.

$$\frac{18x-8}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}$$

Expand \left(x-5\right)\left(x-3\right)\left(x+2\right).

$$\frac{18x-8}{x^{3}-6x^{2}-x+30}$$

Expand

$$\frac{2\left(9x-4\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}$$

Short Solution Steps

Factor x^{2}-8x+15. Factor x^{2}-x-6.

$$\frac{2x+4}{\left(x-5\right)\left(x-3\right)}-\frac{x-5}{\left(x-3\right)\left(x+2\right)}-\frac{x+3}{x^{2}-3x-10}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x-3\right) and \left(x-3\right)\left(x+2\right) is \left(x-5\right)\left(x-3\right)\left(x+2\right). Multiply \frac{2x+4}{\left(x-5\right)\left(x-3\right)} times \frac{x+2}{x+2}. Multiply \frac{x-5}{\left(x-3\right)\left(x+2\right)} times \frac{x-5}{x-5}.

$$\frac{\left(2x+4\right)\left(x+2\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{\left(x-5\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{x+3}{x^{2}-3x-10}$$

Since \frac{\left(2x+4\right)\left(x+2\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)} and \frac{\left(x-5\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x+4\right)\left(x+2\right)-\left(x-5\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{x+3}{x^{2}-3x-10}$$

Do the multiplications in \left(2x+4\right)\left(x+2\right)-\left(x-5\right)\left(x-5\right).

$$\frac{2x^{2}+4x+4x+8-x^{2}+5x+5x-25}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{x+3}{x^{2}-3x-10}$$

Combine like terms in 2x^{2}+4x+4x+8-x^{2}+5x+5x-25.

$$\frac{x^{2}+18x-17}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{x+3}{x^{2}-3x-10}$$

Factor x^{2}-3x-10.

$$\frac{x^{2}+18x-17}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{x+3}{\left(x-5\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x-3\right)\left(x+2\right) and \left(x-5\right)\left(x+2\right) is \left(x-5\right)\left(x-3\right)\left(x+2\right). Multiply \frac{x+3}{\left(x-5\right)\left(x+2\right)} times \frac{x-3}{x-3}.

$$\frac{x^{2}+18x-17}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}-\frac{\left(x+3\right)\left(x-3\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}$$

Since \frac{x^{2}+18x-17}{\left(x-5\right)\left(x-3\right)\left(x+2\right)} and \frac{\left(x+3\right)\left(x-3\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+18x-17-\left(x+3\right)\left(x-3\right)}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}$$

Do the multiplications in x^{2}+18x-17-\left(x+3\right)\left(x-3\right).

$$\frac{x^{2}+18x-17-x^{2}+3x-3x+9}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}$$

Combine like terms in x^{2}+18x-17-x^{2}+3x-3x+9.

$$\frac{18x-8}{\left(x-5\right)\left(x-3\right)\left(x+2\right)}$$

Expand \left(x-5\right)\left(x-3\right)\left(x+2\right).

$$\frac{18x-8}{x^{3}-6x^{2}-x+30}$$