Find the Value

\[ a+b+c=9 \]

\[ ab+bc+ca=23 \]

Find:

\[ a^2+b^2+c^2 \]

Solution:

Using identity:

\[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \]

\[ (9)^2 = a^2+b^2+c^2+2(23) \]

\[ 81 = a^2+b^2+c^2+46 \]

\[ a^2+b^2+c^2 = 81-46 \]

\[ =35 \]

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