Question

若关于x的方程x^2-x+a=0和 x^2-x+b=0(a不等于b)的4个实数根可以组成首项为1/4的等差数列,求|a-

Answer 1

x^2-x+a=0 实数根x1,x2 x^2-x+b=0 实数根x3,x4 4个实数根组成首项为1/4的等差数列:1/4, 1/4+d, 1/4+2d, 1/4+3d x1+x2=x3+x4=1 x1=1/4 ,x2=1/4+3d ,x3=1/4+d ,x4=1/4+2d 1/4+1/4+3d=1 3d=1/2 d=1/6 x1x2=(1/4)*(1/4+1/2)=3/16=-a x3x4=(1/4+1/6)(1/4+1/3)=(5/12)(7/12)=35/144=-b |a-b|=|x3x4-x1x2|=|35/144-3/16|=8/144=1/18