If $a + b + c = 10$ and $a^{2} + b^{2} + c^{2} = 38$ , find $a b + b c + c a$

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Solution

Formula ${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)$
Substituting $a + b + c = 10; a^{2} + b^{2} + c^{2} = 38$
$10^{2} = 38 + 2 (a b + b c + c a)$
$100 - 38 = 2 (a b + b c + c a)$
$a b + b c + c a = \frac{62}{2} = 31$

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