Từ $a+b+c = 0 \Rightarrow b+c =-a \Rightarrow (b+c)^2 =a^2$
$b^2 +c^2 +2bc = a^2 \Rightarrow a^2 -b^2-c^2 =2bc$
$\Rightarrow (a^2-b^2-c^2)^2 =4b^2 c^2$
$\Rightarrow a^4 +b^4 +c^4 -2a^2b^2 +2b^2c^2 -2c^2 a^2 =4b^2c^2$
$\Rightarrow a^4 +b^4 +c^4 = 2a^2b^2 +2b^2c^2 +2a^2c^2$
$ \Rightarrow 2(a^4 +b^4 +c^4) = a^4 +b^4 +c^4 +2a^2b^2 +2b^2c^2 +2a^2c^2 = (a^2 +b^2 +c^2)^2$
Vậy $a^2 +b^2 +c^2 =\sqrt{2(a^4 +b^4 +c^4)}$