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Given that $a(a+2b) = \frac{104}3$, $b(b+2c) = \frac{7}{9}$, and $c(c+2a) = -7$, find $|a+b+c|$.

 May 13, 2018
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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

 May 13, 2018

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