If $a + b + c = 11$ and $a b + b c + a c = 32$
find
$a^{2} + b^{2} + c^{2}$

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Solution

The correct option is A
57

Given $(a + b + c) = 11$ and $(a b + b c + c a) = 32$
$\Rightarrow (a + b + c)^{2} = 11^{2}$
$\Rightarrow a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) = 121$
$\Rightarrow a^{2} + b^{2} + c^{2} + 2 (32) = 121$
$\Rightarrow a^{2} + b^{2} + c^{2} + 64 = 121$
$\Rightarrow a^{2} + b^{2} + c^{2} = 121 - 64 = 57$

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