分解因式
(1)4(x-y+1)+y(y-2x) =y^2-2xy+x^2 +4(x-y)+4-x^2
=(x-y)^2+4(x-y)+4 -x^2= (x-y+2)^2-x^2
=(x-y+2+x)(x-y+2-x)=(2x-y+2)(2-y)
(2)(x^2-2x)^2-7(x^2-2x)+12
=(x^2-2x-3)(x^2-2x-4)=(x-3)(x+1)(x^2-2x-4)
解答题
1.正数x,y满足x^2-y^2=2xy。求x-y/x+y
x^2-2xy+y^2=2y^2, (x-y)^2=2y^2,
x>0,y>0, 2xy>0, x^2-y^2>0, x>y, x-y>0
x-y=√2y,
x+y=√2y+2y=(2+√2)y
(x-y)/(x+y)=√2y/[(2+√2)y]=√2/(2+√2)=√2(2-√2)/2=√2-1
2.已知x+y=1,求x^3+y^3+3xy
x^3+y^3+3xy=(x+y)(x^2-xy+y^2)+3xy=x^2-xy+y^2+3xy
=x^2+2xy+y^2=(x+y)^2=1
3.若b=根号a^2-1+根号1-a^2/a+1,求a+b的值
a^2-1≥0, 1-a^2≥0, 所以 a^2-1=0, a^2=1,
又a+1不等于0, 所以a=1,
a+b=1+0=1
4.已知关于x的不等式2x^2+bx-c>0的解为x<-1,或x>3,解关于x的不等式bx^2+cx+4大于等于0
令 2x^2+bx-c=0, 则x1=-1, x2=3,
x1+x2=-b/2=2, b=-4, x1*x2=-c/2=-3, c=6
解-4x^2+6x+4≥0, 2x^2-3x-2≤0, (2x+1)(x-2)≤0
-1/2≤x≤2
5.若关于x的不等式(a-2)x^2+2(a-2)x-4<0对一切实数x恒成立,求实数a的取值范围
a-2<0,a<2 图象开口向下.
(a-2)x^2+2(a-2)x-4= (a-2)(x^2+2x+1)-(a-2)-4
=(a-2)(x+1)^2 -a-2, x=-1时,x+1=0,函数有最大值,-a-2
令-a-2<0, a>-2
所以-2<a<2
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