Question

\(\text{Tính }A=x^2\left(x+1\right)-y^2\left(y-1\right)+xy-3xy\left(x-y+1\right)+2019\) \(\text{biết }x-y=\sqrt{12\sqrt{5}+29}\)

Answer 1

\(x-y=\sqrt{29+12\sqrt{5}}=2\sqrt{5}+3\)

\(A=x^3-y^3+x^2+y^2+xy-3xy\left(x-y+1\right)+2019\)

\(=\left(x-y\right)\left(x^2+y^2+xy\right)+x^2+y^2+xy-3xy\left(x-y+1\right)+2019\)

\(=\left(x-y+1\right)\left(x^2+y^2+xy\right)-3xy\left(x-y+1\right)+2019\)

\(=\left(x-y+1\right)\left(x^2+y^2-2xy\right)+2019\)

\(=\left(x-y+1\right)\left(x-y\right)^2+2019\)

\(=\left(4+2\sqrt{5}\right)\left(3+2\sqrt{5}\right)^2+2019\)

\(=2255+106\sqrt{5}\)