Question

cho a+b+c=0 va  a²+b²+c²=1

tinh gia tri bthuc M=a⁴+b⁴+c⁴

Answer 1

Ta có :

( a + b + c )² = a² + b² + c² + 2ab + 2 bc+ 2ac = 0

Mà a² + b ² + c² = 1 

=> 2ab + 2bc + 2ac = - 1

=> ab + bc + ac = \(\frac{-1}{2}\)

=> ( ab + bc + ac ) ² = a²b² + a²c² + b²c ² + 2ab²c + 2ac²b + 2a²bc = \(\left(\frac{-1}{2}\right)^2\)=\(\frac{1}{4}\)

=> a²b² + a²c² + b²c² + 2abc ( a + b +c ) = \(\frac{1}{4}\)

mà a + b + c =  0 => 2abc ( a +b +c ) = 0

=> a²b² + b²c² + c²a² = \(\frac{1}{4}\)

Ta có : ( a² + b² + c² )² = a⁴ + b⁴ + c⁴ + 2 ( a²b² + b²c² + c²a² ) = 1

=> a⁴ +b⁴ + c⁴  + 2. \(\frac{1}{4}\) = 1

=> a⁴ + b⁴ + c⁴ = 1 - \(\frac{1}{2}\)

=> a⁴ + b⁴ + c⁴ = \(\frac{1}{2}\)