If $a^{2} + b^{2} + c^{2} = 35$ and $a b + b c + c a = 23$ , find all possible values of $a + b + c$ .

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Solution

Using the identity is $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$

It is given that $a^{2} + b^{2} + c^{2} = 35$ and $a b + b c + c a = 23$ , therefore,

$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) = 35 + (2 \times 23) = 35 + 46 = 81 \Rightarrow (a + b + c)^{2} = 81 \Rightarrow a + b + c = - \sqrt{81}, \sqrt{81} \Rightarrow a + b + c = - 9, 9$

Hence, the value of $(a + b + c)$ is $- 9$ or $9$ .

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