Solve $LaTeX: \displaystyle \frac{x}{x + 9} + \frac{3}{x + 6}=\frac{9}{x^{2} + 15 x + 54}$ .

Factoring the denominator on the right hand side gives $LaTeX: \displaystyle \left(x + 6\right) \left(x + 9\right)$ . This gives the LCD as $LaTeX: \displaystyle \left(x + 6\right) \left(x + 9\right)$ . Multiplying by the LCD gives $LaTeX: \displaystyle x \left(x + 6\right) + 3 x + 27 = 9$ . Getting zero on one side gives $LaTeX: \displaystyle x^{2} + 9 x + 18=0$ . Factoring gives $LaTeX: \displaystyle \left(x + 3\right) \left(x + 6\right)=0$ . The two possible solutions are $LaTeX: \displaystyle x = -3$ and $LaTeX: \displaystyle x = -6$ . Checking the possible solutions gives:
Since $LaTeX: \displaystyle -6$ is zero of the denominator it is not in the domain and must be rejected as a solution. Since $LaTeX: \displaystyle -3$ is not zero of the denominator it is a solution.