Solve x+2/x^2+4x-45+x+7/x^2+6x-27

$$\frac { x + 2 } { x ^ { 2 } + 4 x - 45 } + \frac { x + 7 } { x ^ { 2 } + 6 x - 27 }$$

Evaluate

$$\frac{2x^{2}+x-41}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}$$

Short Solution Steps

Factor x^{2}+4x-45. Factor x^{2}+6x-27.

$$\frac{x+2}{\left(x-5\right)\left(x+9\right)}+\frac{x+7}{\left(x-3\right)\left(x+9\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x+9\right) and \left(x-3\right)\left(x+9\right) is \left(x-5\right)\left(x-3\right)\left(x+9\right). Multiply \frac{x+2}{\left(x-5\right)\left(x+9\right)} times \frac{x-3}{x-3}. Multiply \frac{x+7}{\left(x-3\right)\left(x+9\right)} times \frac{x-5}{x-5}.

$$\frac{\left(x+2\right)\left(x-3\right)}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}+\frac{\left(x+7\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}$$

Since \frac{\left(x+2\right)\left(x-3\right)}{\left(x-5\right)\left(x-3\right)\left(x+9\right)} and \frac{\left(x+7\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+2\right)\left(x-3\right)+\left(x+7\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}$$

Do the multiplications in \left(x+2\right)\left(x-3\right)+\left(x+7\right)\left(x-5\right).

$$\frac{x^{2}-3x+2x-6+x^{2}-5x+7x-35}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}$$

Combine like terms in x^{2}-3x+2x-6+x^{2}-5x+7x-35.

$$\frac{2x^{2}+x-41}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}$$

Expand \left(x-5\right)\left(x-3\right)\left(x+9\right).

$$\frac{2x^{2}+x-41}{x^{3}+x^{2}-57x+135}$$

Expand

$$\frac{2x^{2}+x-41}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}$$

Short Solution Steps

Factor x^{2}+4x-45. Factor x^{2}+6x-27.

$$\frac{x+2}{\left(x-5\right)\left(x+9\right)}+\frac{x+7}{\left(x-3\right)\left(x+9\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x+9\right) and \left(x-3\right)\left(x+9\right) is \left(x-5\right)\left(x-3\right)\left(x+9\right). Multiply \frac{x+2}{\left(x-5\right)\left(x+9\right)} times \frac{x-3}{x-3}. Multiply \frac{x+7}{\left(x-3\right)\left(x+9\right)} times \frac{x-5}{x-5}.

$$\frac{\left(x+2\right)\left(x-3\right)}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}+\frac{\left(x+7\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}$$

Since \frac{\left(x+2\right)\left(x-3\right)}{\left(x-5\right)\left(x-3\right)\left(x+9\right)} and \frac{\left(x+7\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+2\right)\left(x-3\right)+\left(x+7\right)\left(x-5\right)}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}$$

Do the multiplications in \left(x+2\right)\left(x-3\right)+\left(x+7\right)\left(x-5\right).

$$\frac{x^{2}-3x+2x-6+x^{2}-5x+7x-35}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}$$

Combine like terms in x^{2}-3x+2x-6+x^{2}-5x+7x-35.

$$\frac{2x^{2}+x-41}{\left(x-5\right)\left(x-3\right)\left(x+9\right)}$$

Expand \left(x-5\right)\left(x-3\right)\left(x+9\right).

$$\frac{2x^{2}+x-41}{x^{3}+x^{2}-57x+135}$$