Question

[Lương Thanh Tùng]

Cho a + b + c = 0 chứng minh rằng a^2 + b^2 + c^2 = 2( a^4 + b^4 + c^4 )

Answer 1

[Dương Anh]

co a+b+c=0 =>b+c=-a

suy ra (b+c)²=(-a)²  hay b²+2bc+c² =a²

hay b²+c²-a² =-2bc

Suy ra (b² + c² - a² )² =( -2bc)²

<=> b⁴ +c⁴ +a⁴ +2b²c² -2a²b² -2a²c² = 4b²c²

<=> a⁴+b⁴+c⁴ =2a²b²+2b²c²+2c²a²

<=> 2(a⁴+b⁴+c⁴) = a⁴+b⁴+c⁴+2a²b²+2b²c²+2c²a²

<=> a²+b²+c² =2(a⁴+b⁴+c⁴) (dpcm)