How do you solve  5/(x+9) + 11 /( x+2) =9 /( x ^2 + 11x + 18)?

Oct 9, 2015

Solution: $x = - \frac{25}{4}$.

Explanation:

First of all, compute the GCD in the first member:

$\frac{5}{x + 9} + \frac{11}{x + 2} = \frac{5 \left(x + 2\right) + 11 \left(x + 9\right)}{\left(x + 9\right) \left(x + 2\right)}$

Simplifying the numerator, we obtain

$5 x + 10 + 11 x + 99 = 16 x + 109$

Simplifying the denominator, we obtain

${x}^{2} + 2 x + 9 x + 18 = {x}^{2} + 11 x + 18$

So, our equation becomes

$\frac{16 x + 109}{{x}^{2} + 11 x + 18} = \frac{9}{{x}^{2} + 11 x + 18}$

Since the denominators are equal, the equality holds if and only if it holds between the numerators, i.e.

$16 x + 109 = 9 \setminus \iff 16 x = - 100 \setminus \iff x = - \frac{100}{16} = - \frac{25}{4}$

P.S.: the denominator(s) are zero for $x = - 2$ or $x = - 9$, so the root we found is acceptable.