Question

试说明代数式-X的平方-Y的平方+2X-4Y-17的值恒为负数

Answer 1

-x²-y²+2x-4y-17

=(-x²+2x-1)+(-y²-4y-4)-12

=-(x²-2x+1)-(y²+4y+4) -12

=-(x-1)²-(y+2)²-12

=-[(x-1)²+(y-2)²]-12

平方项恒非负,(x-1)²≥0 (y-2)²≥0

(x-1)²+(y-2)²≥0

-[(x-1)²+(y-2)²]≤0

-[(x-1)²+(y-2)²]-12≤-12