If $a + b + c = 7$ and $a b + b c + c a = 20$ find the value of $a^{2} + b^{2} + c^{2}$

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Solution

Given: $a + b + c = 7$ and $a b + b c + c a = 20$
Using the formula, $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 a c$
$\Rightarrow (a + b + c) ² - 2 a b - 2 b c - 2 a c = a^{2} + b^{2} + c^{2}$
$\Rightarrow (7)^{2} - 2 (a b + b c + a c) = a^{2} + b^{2} + c^{2}$
$\Rightarrow 49 - 2 (20) = a^{2} + b^{2} + c^{2}$
$∴ a^{2} + b^{2} + c^{2} = 9$

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