Question

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

CMR: \(a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4\)

Answer 1

Ta có: \(a^2+b^2+\left(a+b\right)^2=c^2+d^2+\left(c+d\right)^2\)

=> \(a^2+b^2+a^2+b^2+2ab=c^2+d^2+c^2+d^2+2cd\)

=> \(a^2+b^2+ab=c^2+d^2+cd\)

=> \(\left(a^2+b^2+ab\right)^2=\left(c^2+d^2+cd\right)^2\)

=> \(a^4+b^4+a^2b^2+2a^2b^2+2a^3b+2b^3a=c^4+d^4+c^2d^2\)

\(+2c^2d^2+2c^3d+2cd^3\)

=> \(2a^4+2b^4+6a^2b^2+4a^3b+4ab^3=2c^4+2d^4+6c^2d^2\)

\(+4c^3d+4cd^3\)

=> \(a^4+b^4+\left(a+b\right)^4=c^4+d^4+\left(c+d\right)^4\)

=> đpcm