Given that $a(a+2b) = \frac{104}3$, $b(b+2c) = \frac{7}{9}$, and $c(c+2a) = -7$, find $|a+b+c|$.
+983 Given that \(a(a+2b) = \frac{104}3, b(b+2c) = \frac{7}{9}, \text{and}\ c(c+2a) = -7, \text{find}\ |a+b+c|.\)
We distribute the variables:
\(a^2+2ab=\frac{104}3,b^2+2bc=\frac79, \text{and} \ c^2+2ac=-7\)
Add the three equations:
\(a^2+b^2+c^2+2ab+2bc+2ac=\frac{104}{3}+\frac79-7\)
We recognize the left side of the equation as the expanded form of \((a+b+c)^2\)
We can rewrite the equation in the form like this:
\((a+b+c)^2=\frac{256}9 \)
Since the problem asks for \( |a+b+c|\), we take the square root:
\(\sqrt{(a+b+c)^2}=\sqrt{\frac{256}9}\\ a+b+c=\pm\frac{16}3 \)
Therefore, \( |a+b+c|=\boxed{\frac{16}3}\)
I hope this helped,
Gavin
