论文部分内容阅读
考察实系数一元二次方程 x~2+px+q=0 (Ⅰ)和圆x~2+y~2+px-(1+q)y+q=0.(Ⅱ)很明显,如果方程(Ⅰ)有实根,那么这些根必定是圆(Ⅱ)与x轴的公共点的横坐标;反之,如果圆(Ⅱ)与x轴有公共点,那么这些点的横坐标必定是方程(Ⅰ)的实根。由于方程(Ⅰ)和圆(Ⅱ)之间存在着这样的关系,我们就可以利用圆(Ⅱ)的几何性质,来研究方程(Ⅰ). 应用配方法,(Ⅱ)式可以改写成
Examine the real coefficient quadratic equation x~2+px+q=0 (I) and circle x~2+y~2+px-(1+q)y+q=0.(II) It is obvious if the equation (I) There are real roots, then these roots must be the abscissa of the common point of the circle (II) and the x-axis; conversely, if the circle (II) and the x-axis have common points, then the abscissa of these points must be the equation ( I) The real root. Due to the existence of such a relationship between equation (I) and circle (II), we can use the geometric properties of circle (II) to study equation (I). With the application method, (II) can be rewritten as