If a + b + c = 0 and $a^{2} + b^{2} + c^{2} = 16$ , find the value of ab + bc + ca.

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Solution

a + b + c = 0

Squaring both sides,

$(a + b + c)^{2} = 0 \Rightarrow a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) = 0 \Rightarrow 16 + 2 (a b + b c + c) = 0 \Rightarrow 2 (a b + b c + c a) = - 16 \Rightarrow a b + b c + c a = \frac{- 16}{2} = - 8 ∴ a b + b c + c a = - 8$

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If $a^{2} + b^{2} + c^{2} = 16$ and ab + bc + ca = 10, find the value of a + b + c.

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