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# Help!

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

Guest 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:

$$a^2+2ab=\frac{104}3,b^2+2bc=\frac79, \text{and} \ c^2+2ac=-7$$

$$a^2+b^2+c^2+2ab+2bc+2ac=\frac{104}{3}+\frac79-7$$

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

GYanggg  May 13, 2018