RE: Exact real solutions of cubic equations

• To: mathgroup at smc.vnet.net
• Subject: [mg46953] RE: [mg46932] Exact real solutions of cubic equations
• From: "David Park" <djmp at earthlink.net>
• Date: Wed, 17 Mar 2004 02:29:07 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Jonas,

First solve a particular case of your equation.

sols = With[
{a = -4, b = 3},
Solve[1 + a s + b s^2 + s^3 == 0, s]];

Extract the actual solutions as sols2 and look at their approximate values.

sols2 = s /. sols;
sols2 // N // Chop
{0.692021, -4.04892, 0.356896}

The following gets exact values, although maybe they are not in the form you
were looking for. If we take approximate values we see that they are real
and the same as above.

ComplexExpand[Re[#]] & /@ sols2
% // N
{-1 + Sqrt[7/3]*
Cos[(1/3)*(Pi - ArcTan[1/(3*Sqrt[3])])] +
Sqrt[7/3]*Cos[(1/3)*(-Pi + ArcTan[
1/(3*Sqrt[3])])],
-1 - (1/2)*Sqrt[7/3]*
Cos[(1/3)*(Pi - ArcTan[1/(3*Sqrt[3])])] -
(1/2)*Sqrt[7/3]*
Cos[(1/3)*(-Pi + ArcTan[1/(3*Sqrt[3])])] -
(1/2)*Sqrt[7]*
Sin[(1/3)*(Pi - ArcTan[1/(3*Sqrt[3])])] +
(1/2)*Sqrt[7]*
Sin[(1/3)*(-Pi + ArcTan[1/(3*Sqrt[3])])],
-1 - (1/2)*Sqrt[7/3]*
Cos[(1/3)*(Pi - ArcTan[1/(3*Sqrt[3])])] -
(1/2)*Sqrt[7/3]*
Cos[(1/3)*(-Pi + ArcTan[1/(3*Sqrt[3])])] +
(1/2)*Sqrt[7]*
Sin[(1/3)*(Pi - ArcTan[1/(3*Sqrt[3])])] -
(1/2)*Sqrt[7]*
Sin[(1/3)*(-Pi + ArcTan[1/(3*Sqrt[3])])]}
{0.692021, -4.04892, 0.356896}

David Park

From: JonasB at iui.se [mailto:JonasB at iui.se]
To: mathgroup at smc.vnet.net

Hello,

I would like to find the _exact_ real roots of some cubic polynomials.
Mathematica seems to have problems determining that a root is real

Solve[1 + a s + b s^2 + s^3 == 0, s]

results in three complex solutions for a = -4 and b = 3. FullSimplify does
not help, either it does nothing or it gets stuck, depending on the values
of a and b. I can of course evaluate the solution numerically, but that is
not what I want. Does anyone know of a package that can simplify expressions
with complex numbers?

Jonas

```

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