Phương trình \( \frac{{x + a}}{{b + c}} + \frac{{x + b}}{{a + c}} + \frac{{x + c}}{{a + b}} = - 3\) có nghiệm là

* Hướng dẫn giải

Ta có:

 \(\begin{array}{l} \frac{{x + a}}{{b + c}} + \frac{{x + b}}{{a + c}} + \frac{{x + c}}{{a + b}} = - 3 \Leftrightarrow \frac{{x + a}}{{b + c}} + \frac{{x + b}}{{a + c}} + \frac{{x + c}}{{a + b}} + 3 = 0\\ \to \left( {\frac{{x + a}}{{b + c}} + 1} \right) + \left( {\frac{{x + b}}{{a + c}} + 1} \right) + \left( {\frac{{x + c}}{{a + b}} + 1} \right) = 0\\ \to \frac{{x + a + b + c}}{{b + c}} + \frac{{x + a + b + c}}{{a + c}} + \frac{{x + a + b + c}}{{a + b}} = 0\\ \to (x + a + b + c)\left( {\frac{1}{{b + c}} + \frac{1}{{a + c}} + \frac{1}{{a + b}}} \right) = 0\\ \to x + a + b + c = 0 \to x = - (a + b + c) \end{array}\)