How do you simplify (5x)/(x^2-9)-(4x)/(x^2+5x+6?

Dec 15, 2016

#(x(x + 22))/((x + 2)(x + 3)(x - 3))

Explanation:

First step to solving this problem is to factor the quadratic equations in the denominators of the two fraction:

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

To get these fractions over a common denominator we need to multiply each fraction by the appropriate form of $1$:

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

$\frac{5 {x}^{2} + 10 x}{\left(x + 2\right) \left(x + 3\right) \left(x - 3\right)} - \frac{4 {x}^{2} - 12 x}{\left(x + 2\right) \left(x + 3\right) \left(x - 3\right)}$

$\frac{5 {x}^{2} + 10 x - 4 {x}^{2} + 12 x}{\left(x + 2\right) \left(x + 3\right) \left(x - 3\right)}$

$\frac{{x}^{2} + 22 x}{\left(x + 2\right) \left(x + 3\right) \left(x - 3\right)}$

$\frac{x \left(x + 22\right)}{\left(x + 2\right) \left(x + 3\right) \left(x - 3\right)}$