If $a^{2} + b^{2} + c^{2} = 20$ and $a + b + c = 0$ find $a b + b c + c a + 10$

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Solution

$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
$\Rightarrow (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$
$\Rightarrow (0)^{2} = 20 + 2 (a b + b c + c a)$
$\Rightarrow - 20 = 2 (a b + b c + c a)$
$\Rightarrow a b + b c + c a + 10 = - 10 + 10 = 0$

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