Determine the formula for $a^{2} + b^{2} + c^{2}$

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Solution

STEP 1 : Expanding ${(a + b + c)}^{2}$
We know that ${(a + b + c)}^{2}$ can be expressed as
${(a + b + c)}^{2} = (a + b + c) (a + b + c)$
${(a + b + c)}^{2} = a^{2} + a b + a c + b a + b^{2} + b c + c a + c b + c^{2}$
${(a + b + c)}^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
STEP 2 : Transposing $2 a b + 2 b c + 2 c a$ on L.H.S.
Transposing $2 a b + 2 b c + 2 c a$ on L.H.S. we get
${(a + b + c)}^{2} - 2 a b - 2 b c - 2 c a = a^{2} + b^{2} + c^{2}$
${\Rightarrow a^{2} + b^{2} + c^{2} = (a + b + c)}^{2} - 2 a b - 2 b c - 2 c a$
On further simplification we get
${\Rightarrow a^{2} + b^{2} + c^{2} = (a + b + c)}^{2} - 2 (a b + b c + c a)$
Hence, the formula for $a^{2} + b^{2} + c^{2}$ is ${a^{2} + b^{2} + c^{2} = (a + b + c)}^{2} - 2 (a b + b c + c a)$ .

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