Question

求证:无论x、y取何值,代数式x^2-xy+y^2-2x+5/2的值总为非负数

Answer 1

配方

x²-xy+y²-2x+5/2

=(1/2)(2x²-2xy+2y²-4x+5)

=(1/2)[(x²-2xy+y²)+(x²-4x+4)+y²+1]

=(1/2)[(x-y)²+(x-2)²+y²+1]

∵(x-y)²≥0,(x-2)²≥0,y²≥0

∴(x-y)²+(x-2)²+y²+1≥1

则

x²-xy+y²-2x+5/2=(1/2)[(x-y)²+(x-2)²+y²+1]≥1/2

即代数式x²-xy+y²-2x+5/2的值总为非负数