If $a^{2} + b^{2} + c^{2} = 90$ and $a + b + c = 20$ . Find the value of $a b + b c + c a$

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Solution

Given $a^{2} + b^{2} + c^{2} = 90$
and also $a + b + c = 20$
Squaring on both sides
$(a + b + c)^{2} = 400$
$(a^{2} + b^{2} + c^{2}) + 2 (a b + b c + c a) = 400$
$⟹ 2 (a b + b c + c a) = 400 - 90 = 310 ⟹ a b + b c + c a = 155$

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