Question

cho a+b+c=0 cmr a^4+b^4+c^4=2(ab+bc+ac)^2

Answer 1

\(a+b+c=0\)

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

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

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

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

\(\Leftrightarrow a^4+b^4+c^4+2a^2b^2+2b^2c^2+2a^2c^2=4\left(ab+bc+ac\right)^2\)

\(\Leftrightarrow a^4+b^4+c^4=4\left(ab+bc+ac\right)^2-2a^2b^2-2b^2c^2-2a^2c^2\)

Mà \(\left(ab+bc+ac\right)^2=a^2b^2+b^2c^2+a^2c^2+abc\left(a+b+c\right)\)

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

nên \(a^4+b^4+c^4=4\left(ab+bc+ac\right)^2-2\left(ab+bc+ac\right)^2\)

\(a^4+b^4+c^4=2\left(ab+bc+ac\right)^2\left(đpcm\right)\)

Answer 2

thanks