Question

Cho a+b+c=0

cm a⁴+b⁴+c⁴=1/2(a²+b²+c²)²

Answer 1

từ a+b+c=0 suy ra a=-(b+c)

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

=> \(a^2-b^2-c^2=2bc\)

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

=> \(a^4+b^4+c^4-2a^2b^2+2b^2c^2-2c^2a^2=4b^2c^2\)

=> \(a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2\)

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

=> \(a^4+b^4+c^4=\frac{1}{2}\cdot\left(a^2+b^2+c^2\right)^2\)