Question

cho ba số a,b,c thoả mãn a+B+C=0 và \(a^2+b^2+c^2=2016\).Tính \(A=a^4+b^4+c^4\)

Answer 1

Ta có : \(a^2+b^2+c^2=2016\)

\(\Leftrightarrow\left(a^2+b^2+c^2\right)^2=2016^2\)

\(\Leftrightarrow a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+a^2c^2\right)=2016^2\)

\(\Leftrightarrow a^4+b^4+c^4=2016^2-2\left(a^2b^2+b^2c^2+a^2c^2\right)\)

Lại có : \(a+b+c=0\)

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

\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ac\right)=0\)

\(\Leftrightarrow2016+2\left(ab+bc+ac\right)=0\)

\(\Leftrightarrow2\left(ab+bc+ac\right)=-2016\)

\(\Leftrightarrow ab+bc+ac=-1008\)

\(\Leftrightarrow\left(ab+bc+ac\right)^2=\left(-1008\right)^2\)

\(\Leftrightarrow a^2b^2+b^2c^2+a^2c^2+2a^2bc+2ab^2c+2abc^2=1008^2\)

\(\Leftrightarrow a^2b^2+b^2c^2+a^2c^2+2abc\left(a+b+c\right)=1008^2\)

\(\Leftrightarrow a^2b^2+b^2c^2+a^2c^2=1008^2\)

Nên : \(A=a^4+b^4+c^4=2016^2-2.1008^2=4064251,587\)

Answer 2

bạn làm sai rồi

2016^2 - 2.1008^2 = 2032128