Question:
If
\[ a+b+c=9 \] and \[ ab+bc+ca=23, \] then \[ a^2+b^2+c^2= \]
(a) 35
(b) 58
(c) 127
(d) none of these
Solution:
Using identity:
\[ (a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca) \]
Substituting the given values:
\[ 9^2 = a^2+b^2+c^2+2(23) \]
\[ 81 = a^2+b^2+c^2+46 \]
\[ a^2+b^2+c^2 = 81-46 \]
\[ =35 \]
Hence, the correct answer is:
\[ \boxed{35} \]