If a + b + c = 9 and ab + bc + ca = 23, find the value of $a^{2} + b^{2} + c^{2}$ .

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Solution

$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a) \Rightarrow (9)^{2} = a^{2} + b^{2} + c^{2} + 2 \times 23 \Rightarrow 81 = a^{2} + b^{2} + c^{2} + 46 \Rightarrow a^{2} + b^{2} + c^{2} = 81 - 46 = 36 ∴ a^{2} + b^{2} + c^{2} = 35$

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a^{2} + b^{2} + c^{2} = 35

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Find $a^{2} + b^{2} + c^{2}; i f a + b + c = 9 a n d a b + b c + c a = 24$

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