Solve x+6/x^2-3x-28+x+5/x^2-5x-36

$$\frac { x + 6 } { x ^ { 2 } - 3 x - 28 } + \frac { x + 5 } { x ^ { 2 } - 5 x - 36 }$$

Evaluate

$$\frac{2x^{2}-5x-89}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}$$

Short Solution Steps

Factor x^{2}-3x-28. Factor x^{2}-5x-36.

$$\frac{x+6}{\left(x-7\right)\left(x+4\right)}+\frac{x+5}{\left(x-9\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x+4\right) and \left(x-9\right)\left(x+4\right) is \left(x-9\right)\left(x-7\right)\left(x+4\right). Multiply \frac{x+6}{\left(x-7\right)\left(x+4\right)} times \frac{x-9}{x-9}. Multiply \frac{x+5}{\left(x-9\right)\left(x+4\right)} times \frac{x-7}{x-7}.

$$\frac{\left(x+6\right)\left(x-9\right)}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}+\frac{\left(x+5\right)\left(x-7\right)}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}$$

Since \frac{\left(x+6\right)\left(x-9\right)}{\left(x-9\right)\left(x-7\right)\left(x+4\right)} and \frac{\left(x+5\right)\left(x-7\right)}{\left(x-9\right)\left(x-7\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+6\right)\left(x-9\right)+\left(x+5\right)\left(x-7\right)}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}$$

Do the multiplications in \left(x+6\right)\left(x-9\right)+\left(x+5\right)\left(x-7\right).

$$\frac{x^{2}-9x+6x-54+x^{2}-7x+5x-35}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}$$

Combine like terms in x^{2}-9x+6x-54+x^{2}-7x+5x-35.

$$\frac{2x^{2}-5x-89}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}$$

Expand \left(x-9\right)\left(x-7\right)\left(x+4\right).

$$\frac{2x^{2}-5x-89}{x^{3}-12x^{2}-x+252}$$

Expand

$$\frac{2x^{2}-5x-89}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}$$

Short Solution Steps

Factor x^{2}-3x-28. Factor x^{2}-5x-36.

$$\frac{x+6}{\left(x-7\right)\left(x+4\right)}+\frac{x+5}{\left(x-9\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x+4\right) and \left(x-9\right)\left(x+4\right) is \left(x-9\right)\left(x-7\right)\left(x+4\right). Multiply \frac{x+6}{\left(x-7\right)\left(x+4\right)} times \frac{x-9}{x-9}. Multiply \frac{x+5}{\left(x-9\right)\left(x+4\right)} times \frac{x-7}{x-7}.

$$\frac{\left(x+6\right)\left(x-9\right)}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}+\frac{\left(x+5\right)\left(x-7\right)}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}$$

Since \frac{\left(x+6\right)\left(x-9\right)}{\left(x-9\right)\left(x-7\right)\left(x+4\right)} and \frac{\left(x+5\right)\left(x-7\right)}{\left(x-9\right)\left(x-7\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+6\right)\left(x-9\right)+\left(x+5\right)\left(x-7\right)}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}$$

Do the multiplications in \left(x+6\right)\left(x-9\right)+\left(x+5\right)\left(x-7\right).

$$\frac{x^{2}-9x+6x-54+x^{2}-7x+5x-35}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}$$

Combine like terms in x^{2}-9x+6x-54+x^{2}-7x+5x-35.

$$\frac{2x^{2}-5x-89}{\left(x-9\right)\left(x-7\right)\left(x+4\right)}$$

Expand \left(x-9\right)\left(x-7\right)\left(x+4\right).

$$\frac{2x^{2}-5x-89}{x^{3}-12x^{2}-x+252}$$