27.已知f(x)是二次函数,且f(x+1)+f(x-1)=2x²-4x,求f(x)的解析式,
∵f(x)是二次函数,∴可设f(x)=ax^2+bx+c.
∴f(x+1)=a(x+1)^2+b(x+1)+c=ax^2+2ax+1+bx+b+c.
f(x-1)=a(x-1)^2+b(x-1)+c=ax^2-2ax+1+bx-b+c.
∴f(x+1)+f(x-1)=2ax^2+2+2bx+2c=2ax^2+2bx+2c+2.······①
又f(x+1)+f(x-1)=2x^2-4x.······②
显然,①、②恒等,∴通过比较各项系数,得:a=1、b=-2、c=-1.
∴f(x)=x^2-2x-1.