Find a+ b + c; if $a^{2} + b^{2} + c^{2} = 83 a n d a b + b c + c a = 71$

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Solution

$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a \Rightarrow (a + b + c)^{2} = 83 + 2 (a b + b c + c a) \Rightarrow (a + b + c)^{2} = 83 + 2 \times 71 \Rightarrow (a + b + c)^{2} = 83 + 142 \Rightarrow (a + b + c)^{2} = 225 \Rightarrow a + b + c = \sqrt{225} \Rightarrow a + b + c = 15$ .

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If $(a + b + c)$ is positive and $a^{2} + b^{2} + c^{2} = 83$ and $a b + b c + c a = 71$

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