Question
If a+b+c = 0, a^2+b^2+c^2 = 4
Find the value of a^4+b^4+c^4
Answer
(a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ac
0 = 4+2ab+2bc+2ac
(1)
ab+bc+ac = -2
(2)
(a^2+b^2+c^2)^2 = a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2c^2
Then,
4^2 = a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2c^2
(3)
Taking square of equation (2)
(ab+bc+ac )^2= 4
a^2b^2+b^2c^2+a^2c^2+2abc(a+b+c) = 4
Then
a^2b^2+b^2c^2+a^2c^2 = 4
Substituting to (3) gives
a^4+b^4+c^4 = 16 - 2\cdot 4 =8