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

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Solution

We know that one of the identity is $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + a c)$ and we are given that $a^{2} + b^{2} + c^{2} = 85$ and $a b + b c + a c = 36$ , therefore, we have:

$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + a c) \Rightarrow (a + b + c)^{2} = 85 + (2 \times 36) \Rightarrow (a + b + c)^{2} = 85 + 72 \Rightarrow (a + b + c)^{2} = 157 \Rightarrow a + b + c = \sqrt{157}$

Hence, $a + b + c = \sqrt{157}$ .

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