求实系数一元三次方程根的实用公式
在数学书籍或数学手册中，对一元三次方程求根公式的叙述都是沿用“卡丹公式”，即：对于一元三次方程：![](https://wkrtcs.bdimg.com/rtcs/image?w=93&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"93","h":"24","dataType":"png","c":"word/media/image1.png","imgOriW":93,"imgOriH":24})
设![](https://wkrtcs.bdimg.com/rtcs/image?w=49&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"49","h":"34","dataType":"png","c":"word/media/image2.png","imgOriW":49,"imgOriH":34})
，
则它的三个根的表达式如下：
![](https://wkrtcs.bdimg.com/rtcs/image?w=159&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"159","h":"24","dataType":"png","c":"word/media/image4.png","imgOriW":159,"imgOriH":24})
![](https://wkrtcs.bdimg.com/rtcs/image?w=190&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"190","h":"24","dataType":"png","c":"word/media/image5.png","imgOriW":190,"imgOriH":24})
![](https://wkrtcs.bdimg.com/rtcs/image?w=190&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"190","h":"24","dataType":"png","c":"word/media/image6.png","imgOriW":190,"imgOriH":24})
其中![](https://wkrtcs.bdimg.com/rtcs/image?w=86&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"86","h":"34","dataType":"png","c":"word/media/image7.png","imgOriW":86,"imgOriH":34})
，
我们先用该公式解一个一元三次方程：![](https://wkrtcs.bdimg.com/rtcs/image?w=92&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"92","h":"24","dataType":"png","c":"word/media/image9.png","imgOriW":91,"imgOriH":24})
。
解： p=- 9，q=6，∴ T=- 3，D=- 18，
∴ 原方程的三个根为
![](https://wkrtcs.bdimg.com/rtcs/image?w=202&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"202","h":"24","dataType":"png","c":"word/media/image10.png","imgOriW":202,"imgOriH":24})
![](https://wkrtcs.bdimg.com/rtcs/image?w=233&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"233","h":"24","dataType":"png","c":"word/media/image11.png","imgOriW":233,"imgOriH":24})
![](https://wkrtcs.bdimg.com/rtcs/image?w=233&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"233","h":"24","dataType":"png","c":"word/media/image12.png","imgOriW":233,"imgOriH":24})
这样求出的三个根的表达式有两个不妥之处：
其一、当![](https://wkrtcs.bdimg.com/rtcs/image?w=37&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"37","h":"17","dataType":"png","c":"word/media/image13.png","imgOriW":37,"imgOriH":17})
时，方程有三个实根（下文给出证明），但这里的、、表达式不明确。
其二、当![](https://wkrtcs.bdimg.com/rtcs/image?w=37&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"37","h":"17","dataType":"png","c":"word/media/image17.png","imgOriW":37,"imgOriH":17})
时，以及（如此例中的）违背了现行中等数学的表示规范，也不能具体地求出其值。
因此，用“卡丹公式”解出的一元三次方程的根，往往是不实用、不直观、不严密的。
下面我们推导一个实用的改进型求根公式。
实系数一元三次方程可写为![](https://wkrtcs.bdimg.com/rtcs/image?w=124&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"124","h":"21","dataType":"png","c":"word/media/image21.png","imgOriW":124,"imgOriH":21})
（1）
令 ![](https://wkrtcs.bdimg.com/rtcs/image?w=60&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"60","h":"33","dataType":"png","c":"word/media/image22.png","imgOriW":60,"imgOriH":33})
，代入（1）得 （2）
其中![](https://wkrtcs.bdimg.com/rtcs/image?w=66&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"66","h":"36","dataType":"png","c":"word/media/image23.png","imgOriW":66,"imgOriH":36})
，
不失一般性，我们只要讨论实系数一元三次方程的求根公式即可。
不妨设p、q均不为零，令y=u+v （3）
代入（2）得，![](https://wkrtcs.bdimg.com/rtcs/image?w=196&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"196","h":"24","dataType":"png","c":"word/media/image25.png","imgOriW":196,"imgOriH":24})
（4）
选择u、v，使得![](https://wkrtcs.bdimg.com/rtcs/image?w=70&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"70","h":"20","dataType":"png","c":"word/media/image26.png","imgOriW":70,"imgOriH":19})
，即 （5）
代入（4）得，![](https://wkrtcs.bdimg.com/rtcs/image?w=76&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"76","h":"24","dataType":"png","c":"word/media/image28.png","imgOriW":76,"imgOriH":24})
（6）
将（5）式两边立方得，![](https://wkrtcs.bdimg.com/rtcs/image?w=72&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"72","h":"37","dataType":"png","c":"word/media/image29.png","imgOriW":72,"imgOriH":37})
（7）
联立（6）、（7）两式，得关于的方程组：
![](https://wkrtcs.bdimg.com/rtcs/image?w=82&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"82","h":"64","dataType":"png","c":"word/media/image30.png","imgOriW":82,"imgOriH":63})
， 且
问题归结于上述方程组的求解。
即求关于t的一元二次方程的两根![](https://wkrtcs.bdimg.com/rtcs/image?w=17&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"17","h":"21","dataType":"png","c":"word/media/image31.png","imgOriW":17,"imgOriH":21})
、，
设![](https://wkrtcs.bdimg.com/rtcs/image?w=82&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"82","h":"37","dataType":"png","c":"word/media/image33.png","imgOriW":82,"imgOriH":37})
，，，
又记的一个立方根为，则另两个立方根为，，
其中![](https://wkrtcs.bdimg.com/rtcs/image?w=17&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"17","h":"21","dataType":"png","c":"word/media/image38.png","imgOriW":17,"imgOriH":21})
，为1的两个立方虚根。
以下分三种情形讨论：
1）若![](https://wkrtcs.bdimg.com/rtcs/image?w=36&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"36","h":"17","dataType":"png","c":"word/media/image40.png","imgOriW":36,"imgOriH":17})
，即D>0，则、均为实数，
可求得![](https://wkrtcs.bdimg.com/rtcs/image?w=78&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"78","h":"21","dataType":"png","c":"word/media/image41.png","imgOriW":78,"imgOriH":21})
，，
取![](https://wkrtcs.bdimg.com/rtcs/image?w=88&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"88","h":"24","dataType":"png","c":"word/media/image43.png","imgOriW":87,"imgOriH":24})
，，
在![](https://wkrtcs.bdimg.com/rtcs/image?w=66&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"66","h":"24","dataType":"png","c":"word/media/image45.png","imgOriW":66,"imgOriH":24})
，组成的九个数中，
有且只有下面三组满足![](https://wkrtcs.bdimg.com/rtcs/image?w=56&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"56","h":"34","dataType":"png","c":"word/media/image27.png","imgOriW":56,"imgOriH":34})
，
即![](https://wkrtcs.bdimg.com/rtcs/image?w=14&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"14","h":"21","dataType":"png","c":"word/media/image35.png","imgOriW":14,"imgOriH":21})
、；、；、，
也就是满足![](https://wkrtcs.bdimg.com/rtcs/image?w=204&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"204","h":"34","dataType":"png","c":"word/media/image52.png","imgOriW":204,"imgOriH":34})
，
∴ 方程（2）的根为![](https://wkrtcs.bdimg.com/rtcs/image?w=66&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"66","h":"21","dataType":"png","c":"word/media/image53.png","imgOriW":66,"imgOriH":21})
，，，
这是方程（2）有一个实根![](https://wkrtcs.bdimg.com/rtcs/image?w=16&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"16","h":"21","dataType":"png","c":"word/media/image14.png","imgOriW":15,"imgOriH":21})
，两个共轭虚根，，
其表达式就是前面给出的“卡丹公式”的形式，
这里的根式![](https://wkrtcs.bdimg.com/rtcs/image?w=28&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"28","h":"20","dataType":"png","c":"word/media/image18.png","imgOriW":28,"imgOriH":19})
及都是在实数意义下的。
2）若![](https://wkrtcs.bdimg.com/rtcs/image?w=36&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"36","h":"17","dataType":"png","c":"word/media/image56.png","imgOriW":36,"imgOriH":17})
，即时，
可求得![](https://wkrtcs.bdimg.com/rtcs/image?w=68&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"68","h":"21","dataType":"png","c":"word/media/image58.png","imgOriW":67,"imgOriH":21})
，取
同理，可求得
![](https://wkrtcs.bdimg.com/rtcs/image?w=121&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"121","h":"21","dataType":"png","c":"word/media/image61.png","imgOriW":121,"imgOriH":21})
∴ 方程（2）有三个实根，其中至少有两个相等的实根。
3）若![](https://wkrtcs.bdimg.com/rtcs/image?w=36&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"36","h":"17","dataType":"png","c":"word/media/image63.png","imgOriW":36,"imgOriH":17})
，即D<0时，
![](https://wkrtcs.bdimg.com/rtcs/image?w=102&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"102","h":"34","dataType":"png","c":"word/media/image64.png","imgOriW":102,"imgOriH":34})
，∴ p<0，，
则、均为虚数，求出、并用三角式表示，
就有![](https://wkrtcs.bdimg.com/rtcs/image?w=94&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"94","h":"21","dataType":"png","c":"word/media/image66.png","imgOriW":94,"imgOriH":21})
，，
其中T，![](https://wkrtcs.bdimg.com/rtcs/image?w=38&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"38","h":"20","dataType":"png","c":"word/media/image68.png","imgOriW":38,"imgOriH":19})
都是实数，
![](https://wkrtcs.bdimg.com/rtcs/image?w=235&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"235","h":"37","dataType":"png","c":"word/media/image69.png","imgOriW":235,"imgOriH":37})
∴ ![](https://wkrtcs.bdimg.com/rtcs/image?w=217&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"217","h":"69","dataType":"png","c":"word/media/image70.png","imgOriW":217,"imgOriH":69})
同理![](https://wkrtcs.bdimg.com/rtcs/image?w=28&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"28","h":"21","dataType":"png","c":"word/media/image72.png","imgOriW":28,"imgOriH":21})
，
其中![](https://wkrtcs.bdimg.com/rtcs/image?w=134&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"134","h":"44","dataType":"png","c":"word/media/image74.png","imgOriW":134,"imgOriH":43})
，且
取![](https://wkrtcs.bdimg.com/rtcs/image?w=169&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"169","h":"37","dataType":"png","c":"word/media/image76.png","imgOriW":169,"imgOriH":37})
，，
则![](https://wkrtcs.bdimg.com/rtcs/image?w=316&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"316","h":"37","dataType":"png","c":"word/media/image78.png","imgOriW":316,"imgOriH":37})
![](https://wkrtcs.bdimg.com/rtcs/image?w=214&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"214","h":"37","dataType":"png","c":"word/media/image79.png","imgOriW":214,"imgOriH":37})
![](https://wkrtcs.bdimg.com/rtcs/image?w=266&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"266","h":"37","dataType":"png","c":"word/media/image80.png","imgOriW":266,"imgOriH":37})
![](https://wkrtcs.bdimg.com/rtcs/image?w=269&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"269","h":"37","dataType":"png","c":"word/media/image81.png","imgOriW":269,"imgOriH":37})
![](https://wkrtcs.bdimg.com/rtcs/image?w=268&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"268","h":"37","dataType":"png","c":"word/media/image82.png","imgOriW":268,"imgOriH":37})
显然，当且仅当取，；，；，
这三组时才满足，
于是方程（2）得三个实根为，，，
具体表示出来就为：
![](https://wkrtcs.bdimg.com/rtcs/image?w=115&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"115","h":"37","dataType":"png","c":"word/media/image85.png","imgOriW":115,"imgOriH":37})
![](https://wkrtcs.bdimg.com/rtcs/image?w=196&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"196","h":"37","dataType":"png","c":"word/media/image86.png","imgOriW":196,"imgOriH":37})
![](https://wkrtcs.bdimg.com/rtcs/image?w=194&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"194","h":"37","dataType":"png","c":"word/media/image87.png","imgOriW":194,"imgOriH":37})
其中![](https://wkrtcs.bdimg.com/rtcs/image?w=134&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"134","h":"44","dataType":"png","c":"word/media/image88.png","imgOriW":134,"imgOriH":43})
∴ 当时，方程（2）有三个实根。
综上所述，实系数一元三次方程的求根公式如下：
令![](https://wkrtcs.bdimg.com/rtcs/image?w=97&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"97","h":"34","dataType":"png","c":"word/media/image3.png","imgOriW":97,"imgOriH":34})
，，，
，
1）当![](https://wkrtcs.bdimg.com/rtcs/image?w=37&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"37","h":"17","dataType":"png","c":"word/media/image89.png","imgOriW":37,"imgOriH":17})
时，方程有一个实根和两个共轭虚根，
2）当![](https://wkrtcs.bdimg.com/rtcs/image?w=37&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"37","h":"17","dataType":"png","c":"word/media/image57.png","imgOriW":37,"imgOriH":17})
时，方程有三个实根，其中至少有两个相等的实根，
![](https://wkrtcs.bdimg.com/rtcs/image?w=65&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"65","h":"22","dataType":"png","c":"word/media/image90.png","imgOriW":65,"imgOriH":22})
， ，
3）当时，方程有三个实根，
，
上面提供的公式，可以求出任意实系数一元三次方程的根的具体值，是实用性的。
这里用以下几个解一元三次方程的实例来说明该公式的应用。
例一、解方程![](https://wkrtcs.bdimg.com/rtcs/image?w=106&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"106","h":"24","dataType":"png","c":"word/media/image92.png","imgOriW":106,"imgOriH":24})
。
解： p=- 27，q=54，∴ D=0，
∴ 原方程的根为![](https://wkrtcs.bdimg.com/rtcs/image?w=81&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"81","h":"22","dataType":"png","c":"word/media/image93.png","imgOriW":81,"imgOriH":22})
，。
例二、解方程![](https://wkrtcs.bdimg.com/rtcs/image?w=92&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"92","h":"24","dataType":"png","c":"word/media/image95.png","imgOriW":91,"imgOriH":24})
。
解： p=9，q=4，∴ D=31>0，
∴ 原方程有一个实根和两个共轭虚根：
![](https://wkrtcs.bdimg.com/rtcs/image?w=182&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"182","h":"25","dataType":"png","c":"word/media/image96.png","imgOriW":182,"imgOriH":25})
![](https://wkrtcs.bdimg.com/rtcs/image?w=325&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"325","h":"34","dataType":"png","c":"word/media/image97.png","imgOriW":325,"imgOriH":34})
![](https://wkrtcs.bdimg.com/rtcs/image?w=402&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"402","h":"34","dataType":"png","c":"word/media/image98.png","imgOriW":402,"imgOriH":34})
![](https://wkrtcs.bdimg.com/rtcs/image?w=446&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"446","h":"34","dataType":"png","c":"word/media/image99.png","imgOriW":446,"imgOriH":34})
例三、解方程。
解： p=- 9，q=6，∴ D=- 18<0，![](https://wkrtcs.bdimg.com/rtcs/image?w=105&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"105","h":"34","dataType":"png","c":"word/media/image100.png","imgOriW":105,"imgOriH":34})
∴ 原方程有三个实根：
![](https://wkrtcs.bdimg.com/rtcs/image?w=176&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"176","h":"34","dataType":"png","c":"word/media/image101.png","imgOriW":176,"imgOriH":34})
![](https://wkrtcs.bdimg.com/rtcs/image?w=334&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"334","h":"34","dataType":"png","c":"word/media/image102.png","imgOriW":334,"imgOriH":34})
![](https://wkrtcs.bdimg.com/rtcs/image?w=331&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"331","h":"34","dataType":"png","c":"word/media/image103.png","imgOriW":331,"imgOriH":34})
通过差表或计算器、计算机可计算得：
![](https://wkrtcs.bdimg.com/rtcs/image?w=108&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"108","h":"21","dataType":"png","c":"word/media/image104.png","imgOriW":108,"imgOriH":21})
![](https://wkrtcs.bdimg.com/rtcs/image?w=118&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"118","h":"21","dataType":"png","c":"word/media/image105.png","imgOriW":118,"imgOriH":21})
![](https://wkrtcs.bdimg.com/rtcs/image?w=115&md5sum=c33588add6469d7e192ebd2de8948433&sign=06825ece10&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=3&ipr={"t":"img","w":"115","h":"21","dataType":"png","c":"word/media/image106.png","imgOriW":115,"imgOriH":21})
这在工程技术上是极其有用的。
本文来源：https://www.2haoxitong.net/k/doc/08847e5f804d2b160b4ec0a8.html
文档为doc格式