If a + b + c = 9 and ab + bc + ca = 40, Find a² + b² + c²
Question:
If \(a+b+c=9\) and \(ab+bc+ca=40\), find \(a^2+b^2+c^2\).
Solution
We know that
\[ (a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca) \]
Given:
\[ a+b+c=9 \]
\[ ab+bc+ca=40 \]
Substituting the values:
\[ 9^2=a^2+b^2+c^2+2(40) \]
\[ 81=a^2+b^2+c^2+80 \]
\[ a^2+b^2+c^2=81-80 \]
\[ a^2+b^2+c^2=1 \]
Therefore,
\[ \boxed{a^2+b^2+c^2=1} \]