Question

Chứng minh : a⁴ + b⁴ + c⁴ = 2( ab + ac + bc )² . Biết a + b + c = 0

Answer 1

a+b+c = 0

=> \(\left(a+b+c\right)^2=0=>a^2+b^2+c^2+2ab+2ac+2bc=0\)

=>\(a^2+b^2+c^2=-2\left(ab+ac+bc\right)\)

bình phương 2 vế ta được

\(a^4+b^4+c^4+2a^2b^2+2a^2c^2+2b^2c^2=4\left(a^2b^2+a^2c^2+b^2c^2+2a^2bc+2ab^2c+2abc^2\right)\left(1\right)\)=> \(a^4+b^4+c^4=4\left[a^2b^2+a^2c^2+b^2c^2+2abc\left(a+b+c\right)\right]-2\left(a^2b^2+a^2c^2+b^2c^2\right)\)=>\(a^4+b^4+c^4=2\left(a^2b^2+a^2c^2+b^2c^2\right)\) (vì a+b+c=0) (2)

từ (1) và (2) => \(2\left(a^4+b^4+c^4\right)=4\left(a^2b^2+a^2c^2+b^2c^2+2a^2bc+2ab^2c+2abc^2\right)\) =>\(a^4+b^4+c^4=2\left(ab+ac+bc\right)^2\)