Find the Value

\[ a+b+c=0 \]

\[ a^2+b^2+c^2=16 \]

Find:

\[ ab+bc+ca \]

Solution:

Using identity:

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

\[ (0)^2 = 16+2(ab+bc+ca) \]

\[ 0 = 16+2(ab+bc+ca) \]

\[ 2(ab+bc+ca) = -16 \]

\[ ab+bc+ca = -8 \]

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