If a² + b² + c² = 20 and a + b + c = 0, Find ab + bc + ca
Question:
If \(a^2+b^2+c^2=20\) and \(a+b+c=0\), find \(ab+bc+ca\).
Solution
We know that
\[ (a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca) \]
Given:
\[ a+b+c=0 \]
Therefore,
\[ (a+b+c)^2=0 \]
Substituting the given values:
\[ 0=20+2(ab+bc+ca) \]
\[ 2(ab+bc+ca)=-20 \]
\[ ab+bc+ca=-10 \]
Therefore,
\[ \boxed{ab+bc+ca=-10} \]