Given that \(a(a+2b) = \frac{104}3\), \(b(b+2c) = \frac{7}{9},\) and \(c(c+2a) = -7,\) find \(|a+b+c|.\)
Rewrite the equations as:
a^2 + 2ab = 104/3
b^2 + 2bc = 7/9
c^2 + 2ac = -7
Add them up:
a^2 + b^2 + c^2 + 2ab + 2bc + 2ac = 256/9
But a^2 + b^2 + c^2 + 2ab + 2bc + 2ac = (a + b + c)^2
so (a + b + c)^2 = 256/9
hence |a + b + c| = 16/3
+109 
