Question

Bài 12: Cho a + b +c = 0 ; a² +b² + c² = 1. Tính giá trị biểu thức a⁴ +b⁴+ c⁴

Answer 1

\(a+b+c=0\Rightarrow\left(a+b+c\right)^2=0\)

\(\Rightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)=0\)

\(\Rightarrow ab+bc+ca=-\dfrac{a^2+b^2+c^2}{2}=-\dfrac{1}{2}\)

\(\Rightarrow\left(ab+bc+ca\right)^2=\dfrac{1}{4}\)

\(\Rightarrow a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)=\dfrac{1}{4}\)

\(\Rightarrow a^2b^2+b^2c^2+c^2a^2=\dfrac{1}{4}\) (do \(a+b+c=0\))

\(\Rightarrow a^4+b^4+c^4=\left(a^2+b^2+c^2\right)^2-2\left(a^2b^2+b^2c^2+c^2a^2\right)=1^2-2.\dfrac{1}{4}=\dfrac{1}{2}\)