Question

cho a+b+c=0 và a²+b²+c²=1

tính A=a⁴+b⁴+c⁴

Answer 1

\(a+b+c=0\\ \Rightarrow\left(a+b+c\right)^2=0\\ \Rightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)=0\\ \Rightarrow ab+bc+ca=-\dfrac{1}{2}\)

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

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