（1）试证明无论x,y为何值,4x^2-12x+9y^2+30y+36的值都为正数.（2）2a^2+4a+2(3)3m^2n^2-18mn+27(4)(x^2+y^2)^2-4x^2y^2(5)4x^2-(x^2+1)^2(6)2(a-1)^2-12(a-1)+18(7)m^2+4m+3

Question

（1）试证明无论x,y为何值,4x^2-12x+9y^2+30y+36的值都为正数.
（2）2a^2+4a+2 (3)3m^2n^2-18mn+27
(4)(x^2+y^2)^2-4x^2y^2 (5)4x^2-(x^2+1)^2
(6)2(a-1)^2-12(a-1)+18 (7)m^2+4m+3

Answer 1

1
4x^2-12x+9y^2+30y+36
=(4x^2-12x+9)+(9y^2+30y+25)+2
=(2x-3)^2+(3y+5)^2+2
可见是正数
2
2a^2+4a+2
=2(a^2+2a+1)
=a(a+1)^2
3
3m^2n^2-18mn+27
=3(m^2n^2-6mn+9)
=3(mn-3)^2
4
(x^2+y^2)^2-4x^2y^2
=[(x^2+y^2)+2xy][(x^2+y^2)-2xy]
=(x+y)^2(x-y)^2
5
4x^2-(x^2+1)^2
=[2x+x^2+1][2x-x^2-1]
=-(x+1)^2(x-1)^2
6
2(a-1)^2-12(a-1)+18
=2[(a-1)^2-6(a-1)+9]
=2[(a-1)-3]^2
=2(a-4)^2
7
m^2+4m+3
=m^2+3m+m+3
=m(m+3)+(m+3)
=(m+3)(m+1)