Solve x+9/x^2-x-30+x+4/x^2+2x-15

$$\frac { x + 9 } { x ^ { 2 } - x - 30 } + \frac { x + 4 } { x ^ { 2 } + 2 x - 15 }$$

Evaluate

$$\frac{2x^{2}+4x-51}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}$$

Short Solution Steps

Factor x^{2}-x-30. Factor x^{2}+2x-15.

$$\frac{x+9}{\left(x-6\right)\left(x+5\right)}+\frac{x+4}{\left(x-3\right)\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-6\right)\left(x+5\right) and \left(x-3\right)\left(x+5\right) is \left(x-6\right)\left(x-3\right)\left(x+5\right). Multiply \frac{x+9}{\left(x-6\right)\left(x+5\right)} times \frac{x-3}{x-3}. Multiply \frac{x+4}{\left(x-3\right)\left(x+5\right)} times \frac{x-6}{x-6}.

$$\frac{\left(x+9\right)\left(x-3\right)}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}+\frac{\left(x+4\right)\left(x-6\right)}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}$$

Since \frac{\left(x+9\right)\left(x-3\right)}{\left(x-6\right)\left(x-3\right)\left(x+5\right)} and \frac{\left(x+4\right)\left(x-6\right)}{\left(x-6\right)\left(x-3\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+9\right)\left(x-3\right)+\left(x+4\right)\left(x-6\right)}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}$$

Do the multiplications in \left(x+9\right)\left(x-3\right)+\left(x+4\right)\left(x-6\right).

$$\frac{x^{2}-3x+9x-27+x^{2}-6x+4x-24}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}$$

Combine like terms in x^{2}-3x+9x-27+x^{2}-6x+4x-24.

$$\frac{2x^{2}+4x-51}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}$$

Expand \left(x-6\right)\left(x-3\right)\left(x+5\right).

$$\frac{2x^{2}+4x-51}{x^{3}-4x^{2}-27x+90}$$

Expand

$$\frac{2x^{2}+4x-51}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}$$

Short Solution Steps

Factor x^{2}-x-30. Factor x^{2}+2x-15.

$$\frac{x+9}{\left(x-6\right)\left(x+5\right)}+\frac{x+4}{\left(x-3\right)\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-6\right)\left(x+5\right) and \left(x-3\right)\left(x+5\right) is \left(x-6\right)\left(x-3\right)\left(x+5\right). Multiply \frac{x+9}{\left(x-6\right)\left(x+5\right)} times \frac{x-3}{x-3}. Multiply \frac{x+4}{\left(x-3\right)\left(x+5\right)} times \frac{x-6}{x-6}.

$$\frac{\left(x+9\right)\left(x-3\right)}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}+\frac{\left(x+4\right)\left(x-6\right)}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}$$

Since \frac{\left(x+9\right)\left(x-3\right)}{\left(x-6\right)\left(x-3\right)\left(x+5\right)} and \frac{\left(x+4\right)\left(x-6\right)}{\left(x-6\right)\left(x-3\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+9\right)\left(x-3\right)+\left(x+4\right)\left(x-6\right)}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}$$

Do the multiplications in \left(x+9\right)\left(x-3\right)+\left(x+4\right)\left(x-6\right).

$$\frac{x^{2}-3x+9x-27+x^{2}-6x+4x-24}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}$$

Combine like terms in x^{2}-3x+9x-27+x^{2}-6x+4x-24.

$$\frac{2x^{2}+4x-51}{\left(x-6\right)\left(x-3\right)\left(x+5\right)}$$

Expand \left(x-6\right)\left(x-3\right)\left(x+5\right).

$$\frac{2x^{2}+4x-51}{x^{3}-4x^{2}-27x+90}$$