怎样求一元三次方程的根

怎样求一元三次方程的根
怎样求一元三次方程的根

怎样求一元三次方程的根
标准型
形如aX^3＋bX^2＋cX＋d=0,（a,b,c,d∈R,且a≠0）的方程是一元三次方程的标准型.
编辑本段公式解法
1．卡尔丹公式法
（卡尔达诺公式法） 特殊型一元三次方程X^3＋pX＋q=0 (p、q∈R) 判别式Δ=(q/2)^2＋(p/3)^3 【卡尔丹公式】 X1=(Y1)^(1/3)＋(Y2)^(1/3)； X2= (Y1)^(1/3)ω＋(Y2)^(1/3)ω^2； 标准型方程中卡尔丹公式的一个实根
X3=(Y1)^(1/3)ω^2＋(Y2)^(1/3)ω,其中ω=(－1＋i3^(1/2))/2； Y(1,2)=－(q/2)±((q/2)^2＋(p/3)^3)^(1/2).标准型一元三次方程aX ^3＋bX ^2＋cX＋d=0 令X=Y—b/(3a)代入上式,可化为适合卡尔丹公式直接求解的特殊型一元三次方程Y^3＋pY＋q=0.【卡尔丹判别法】 当Δ=(q/2)^2＋(p/3)^3>0时,方程有一个实根和一对共轭虚根； 当Δ=(q/2)^2＋(p/3)^3=0时,方程有三个实根,其中有一个两重根； 当Δ=(q/2)^2＋(p/3)^30时,盛金公式②：X⑴=(－b－Y⑴^(1/3)－Y⑵^(1/3))/(3a)； X(2,3)=(－2b＋Y⑴^(1/3)＋Y⑵^(1/3))/(6a)±i3^(1/2)(Y⑴^(1/3)－Y⑵^(1/3))/(6a)； 其中Y(1,2)=Ab＋3a(－B±(B^2－4AC)^(1/2))/2,i^2=－1.当Δ=B^2－4AC=0时,盛金公式③：X⑴=－b/a＋K；X⑵=X3=－K/2,其中K=B/A,(A≠0).当Δ=B^2－4AC0,－1

科学计算器

可以通过求导的方式

求导的方式是可以，不过比较繁琐。你想想，我们求导首先得到的是方程曲线的最低点或者最高点，而不是曲线与X轴的交点（解），还需还进行转换。我有一个VB做的程序你自己做来看看。

在VB中建立text1，text2，text3，text4和text5五个文本框和command1命令按钮。text1，text2，text3，text4为对应的a、b、c、d系数输入框，text5为方程解的...

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求导的方式是可以，不过比较繁琐。你想想，我们求导首先得到的是方程曲线的最低点或者最高点，而不是曲线与X轴的交点（解），还需还进行转换。我有一个VB做的程序你自己做来看看。

在VB中建立text1，text2，text3，text4和text5五个文本框和command1命令按钮。text1，text2，text3，text4为对应的a、b、c、d系数输入框，text5为方程解的输出框，command1按钮为操作按钮（先输入系数再计算）。在代码窗口中输入以下代码：
Private Function cubic(ByVal a As Double, ByVal b As Double, ByVal c As Double, ByVal d As Double) As String
Dim x1r As Double, x1i As Double, x2r As Double, x2i As Double, x3r As Double, x3i As Double
Dim ret As String
Dim J1 As String, J2 As String, J3 As String, J As String
hh = Chr(13) + Chr(10)
ret = CubicEquation(a, b, c, d, x1r, x1i, x2r, x2i, x3r, x3i)
If x1i = 0 Then
J1 = "X1=" & Format$(x1r, "#0.0###############") & ";" + hh
End If
If x1i > 0 Then
J1 = "X1=" & Format$(x1r, "#0.0###############") & " + " & Format$(x1i, "#0.0###############") & " i" & ";" + hh
End If
If x1i < 0 Then
J1 = "X1=" & Format$(x1r, "#0.0###############") & Format$(x1i, "#0.0###############") & " i" & ";" + hh
End If
If x2i = 0 Then
J2 = "X2=" & Format$(x2r, "#0.0###############") & ";" + hh
End If
If x2i > 0 Then
J2 = "X2=" & Format$(x2r, "#0.0###############") & " + " & Format$(x2i, "#0.0###############") & " i" & ";" + hh
End If
If x2i < 0 Then
J2 = "X2=" & Format$(x2r, "#0.0###############") & Format$(x2i, "#0.0###############") & " i" & ";" + hh
End If
If x3i = 0 Then
J3 = "X3=" & Format$(x3r, "#0.0###############") & ";" + hh
End If
If x3i > 0 Then
J3 = "X3=" & Format$(x3r, "#0.0###############") & " + " & Format$(x3i, "#0.0###############") & " i" & ";" + hh
End If
If x3i < 0 Then
J3 = "X3=" & Format$(x3r, "#0.0###############") & Format$(x3i, "#0.0###############") & " i" & ";" + hh
End If
J = J1 + J2 + J3
cubic = J
End Function
Private Function CubicEquation _
(ByVal a As Double, ByVal b As Double, ByVal c As Double, ByVal d As Double, _
x1r As Double, x1i As Double, x2r As Double, x2i As Double, x3r As Double, x3i As Double) As String
Dim e As Double, f As Double, g As Double, h As Double, delta As Double
Dim r As Double, sita As Double, pi As Double, rr As Double, ri As Double
If a = 0 Then
CubicEquation = "Not a cubic equation: a = 0"
Exit Function
End If
'pi = 3.14159265358979
pi = 4 * Atn(1)
b = b / a 'simplify to a=1: x^3+bx^2+cx+d=0
c = c / a
d = d / a
e = -b ^ 2 / 3 + c 'substitute x=y-b/3: y^3+ey+f=0
f = (2 * b ^ 2 - 9 * c) * b / 27 + d
If e = 0 And f = 0 Then
x1r = -b / 3
x2r = x1r
x3r = x1r
CubicEquation = "3 same real roots:"
ElseIf e = 0 Then 'need to deal with e = 0, or it will cause z = 0 later.
r = -f 'y^3+f=0, y^3=-f
r = Cur(r)
x1r = r - b / 3 'a real root
If r > 0 Then 'r never = 0 since g=f/2, f never = 0 there
sita = 2 * pi / 3
x2r = r * Cos(sita) - b / 3
x2i = r * Sin(sita)
Else
sita = pi / 3
x2r = -r * Cos(sita) - b / 3
x2i = -r * Sin(sita)
End If
x3r = x2r
x3i = -x2i
CubicEquation = "1 real root and 2 image roots:"
Else 'substitute y=z-e/3/z: (z^3)^2+fz^3-(e/3)^3=0, z^3=-g+sqr(delta)
g = f / 2 '-q-sqr(delta) is ignored
h = e / 3
delta = g ^ 2 + h ^ 3
If delta < 0 Then
r = Sqr(g ^ 2 - delta)
sita = Argument(-g, Sqr(-delta)) 'z^3=r(con(sita)+isin(sita))
r = Cur(r)
rr = r - h / r
sita = sita / 3 'z1=r(cos(sita)+isin(sita))
x1r = rr * Cos(sita) - b / 3 'y1=(r-h/r)cos(sita)+i(r+h/r)sin(sita), x1=y1-b/3
sita = sita + 2 * pi / 3 'no image part since r+h/r = 0
x2r = rr * Cos(sita) - b / 3
sita = sita + 2 * pi / 3
x3r = rr * Cos(sita) - b / 3
CubicEquation = "3 real roots:"
Else 'delta >= 0
r = -g + Sqr(delta)
r = Cur(r)
rr = r - h / r
ri = r + h / r
If ri = 0 Then
CubicEquation = "3 real roots:"
Else
CubicEquation = "1 real root and 2 image roots:"
End If
x1r = rr - b / 3 'a real root
If r > 0 Then 'r never = 0 since g=f/2, f never = 0 there
sita = 2 * pi / 3
x2r = rr * Cos(sita) - b / 3
x2i = ri * Sin(sita)
Else 'r < 0
sita = pi / 3
x2r = -rr * Cos(sita) - b / 3
x2i = -ri * Sin(sita)
End If
x3r = x2r
x3i = -x2i
End If
End If
End Function
Private Function Cur(v As Double) As Double
If v < 0 Then
Cur = -(-v) ^ (1 / 3)
Else
Cur = v ^ (1 / 3)
End If
End Function
Private Function Argument(a As Double, b As Double) As Double
Dim sita As Double, pi As Double
'pi = 3.14159265358979
pi = 4 * Atn(1)
If a = 0 Then
If b >= 0 Then
Argument = pi / 2
Else
Argument = -pi / 2
End If
Else
sita = Atn(Abs(b / a))

If a > 0 Then
If b >= 0 Then
Argument = sita
Else
Argument = -sita
End If
ElseIf a < 0 Then
If b >= 0 Then
Argument = pi - sita
Else
Argument = pi + sita
End If
End If
End If
End Function
Private Sub Command1_Click()
Dim a As Double, b As Double, c As Double, d As Double
Dim J As String, J1 As String, J2 As String, P As Double
Dim xr As Double, xi As Double
hh = Chr(13) + Chr(10)
a = Val(Text1.Text)
b = Val(Text2.Text)
c = Val(Text3.Text)
d = Val(Text4.Text)
If a <> 0 Then
Text5.Text = cubic(a, b, c, d)
End If
If a = 0 And b <> 0 Then
P = c ^ 2 - 4 * b * d
xr = -c / (2 * b)
Select Case P
Case Is = 0
x = xr
J = "X=" & Format$(x, "#0.0###############")
Text5.Text = J
Case Is > 0
xi = Sqr(Abs(P)) / (2 * b)
J1 = xr + Sqr(Abs(P)) / (2 * b)
J2 = xr - Sqr(Abs(P)) / (2 * b)
J = "X1=" & Format$(J1, "#0.0###############") & hh & "X2=" & Format$(J2, "#0.0###############")
Text5.Text = J
Case Is < 0
xi = Sqr(Abs(P)) / (2 * b)
J1 = "X1=" & Format$(xr, "#0.0###############") & "+" & Format$(xi, "#0.0###############") & "i;" + hh
J2 = "X2=" & Format$(xr, "#0.0###############") & "-" & Format$(xi, "#0.0###############") & "i;"
J = J1 + J2
Text5.Text = J
End Select
End If
If a = 0 And b = 0 And c <> 0 Then
x = d / c
J = "X=" & Format$(x, "#0.0###############")
Text5.Text = J
End If
If a = 0 And b = 0 And c = 0 Then
MsgBox "方程无意义，请重新输入！", , "温馨提示"
End If
End Sub

收起

你想问什么阶段的？高中？大学？
1.高等数学里是有一节内容是讲如何利用求导计算高阶方程的近似解。
2.如果是高中阶段的话，我相信还是要利用因式分解的方法。