Question

cho \(a=\sqrt{2013}+\sqrt{2014}\)

rút gọn :::   \(A=a^4-8054a^2+2015\)

Answer 1

đặt \(\sqrt{2013}=x;\sqrt{2014}=y\)

=>A=\(\left(x+y\right)^4-2.\left(2013+2014\right)\left(x+y\right)^2+2015\)

=>A=\(\left(x+y\right)^4-2.\left(x^2+y^2\right)\left(x+y\right)^2+\left(x^2+y^2\right)^2-\left(x^2+y^2\right)^2\)

=>A=\(\left(x^2+2xy+y^2-x^2-y^2\right)-x^4-2x^2y^2-y^4+2015\)

=>A=\(4x^2y^2-x^4-2x^2y^2-y^4+2015=2015-\left(x^2-y^2\right)^2=2015-\left(2013-2014\right)^2=2014\)

Answer 2

Đặt căn 2013=x ; căn 2014 =y
=> 2(x²+y²)=8054; y²-x²=1
thay vào đc pt: (x+y)⁴-2(x²+y²)(x+y)²+2015
=(x+y)²(x²+2xy+y²-2x²-2y²)+2015
=-(x+y)²(x-y)²+2015
=-(x²-y²)²+2015
=-1+2015 (vì y²-x²=1)
=2014