Question

\(\text{a4 +b4+c4=(a2+b2+c2)^2/2}\)

cho a+b+c=0

Answer 1

Từ a + b + c =0 => -a = -(b + c) => a² = (b + c)²

<=> a² - b² - c² = 2bc

<=> (a² - b² - c²)² = 4b²c²

<=> a⁴ + b⁴ + c⁴ - 2a²b² + 2b²c² - 2c²a² = 4b²c²

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

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

<=> 2(a⁴ + b⁴ + c⁴) = (a² + b² + c²)²

<=> a⁴ + b⁴ + c⁴ = \(\frac{\left(a^2+b^2+c^2\right)^2}{2}\) (đpcm)