If $a b + b c + c a = 36$ and $a^{2} + b^{2} + c^{2} = 85$ then find $a + b + c$ .

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Solution

Consider, $a b + b c + c a = 36$
Multiplying, both sides by $2$ , we get,
$\Rightarrow 2 (a b + b c + c a) = 72$ ..(i)
Consider, $a^{2} + b^{2} + c^{2} = 85$ ...(ii)
Add (i) and (ii), we get,
$\Rightarrow a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) = 157$
$\Rightarrow (a + b + c)^{2} = 157$
$∴ a + b + c = \sqrt{157}$

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