If a+b+c = 10 and $a^{2} + b^{2} + c^{2} = 38; f i n d a b + b c + c a$

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Solution

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

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