Question

Cho a+b+c=0 CMR

\(\left(a^2+b^2+c^2\right)=2\left(a^4+b^4+c^4\right)\)

Answer 1

\(a+b+c=0\)

\(\Rightarrow a^2+b^2+c^2\)= \(2ab+2bc+2ca=0\)

\(\Rightarrow a^2+b^2+c^2\)= \(-2\left(ab+bc+ca\right)\)

\(\Rightarrow\left(a^2+b^2+c^2\right)^2\)= \(\left(-2ab-2bc-2ca\right)^2\)

\(\Rightarrow a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2a^2\)= \(4a^2b^2+4b^2c^2+4c^2a^2+4abc\left(a+b+c\right)\)= \(4a^2b^2+4b^2c^2+4c^2a^2\)( Do a + b + c = 0 )

\(\Rightarrow a^4+b^4+c^4\)= \(2\left(a^2b^2+b^2c^2+c^2a^2\right)\).